Patent Application
20200276493
The present technology relates to the field of games, and more particularly to strategy games.
A portion of the disclosure of this patent document contains material which is subject to copyright protection. The copyright owner has no objection to the facsimile reproduction by anyone of the patent disclosure, as it appears in the Patent and Trademark Office patent files or records, but otherwise reserves all copyright rights whatsoever.
Strategy games of the present technology may be played by at least one player. The playing surface comprises a region that includes a bounded shape having an area divided into a plurality of sectors. Each sector comprises a bounded shape having an area within the region that does not overlap with any other sector. Each sector contains a set of elements, each element of the set of elements having a type and quantity. During play, each player makes one move per turn, according to a set of rules defining types of moves that can be made by the at least one player and restrictions governing how districts can be formed from the plurality of sectors, in pursuit of combining the plurality of sectors into a given number of districts in a manner that seeks to achieve a pre-defined goal based on an aggregation of the elements within each district.
Methods of playing strategy games of the present technology include providing a playing surface that includes a region comprising a bounded shape having an area divided into a plurality of sectors. Each sector comprises a bounded shape having an area within the region that does not overlap with any other sector, and each sector contains a set of elements, each element of the set of elements having a type and quantity. Methods of playing strategy games of the present technology further include making one move per turn per player, according to a set of rules defining types of moves that can be made by the at least one player and restrictions governing how districts can be formed from the plurality of sectors, in pursuit of combining the plurality of sectors into a given number of districts in a manner that seeks to achieve a pre-defined goal based on an aggregation of the elements within each district.
Specific examples have been chosen for purposes of illustration and description, and are shown in the accompanying drawings, forming a part of the specification. Within the Figures, like parts have been given like numbers for ease of reference. It should be understood that the drawings are not necessarily drawn to scale and that they are intended to be merely illustrative.
The present technology provides strategy games involving the division of a region into districts to achieve one or more predefined goals. The use of the prefix “pre” herein means any time prior to beginning the play of the game. Games of the present invention can be provided as board games, sector based games, paper based games, electronic games, or games on any other suitable presentation medium. For example, the playing surface on which a region may be provided may take the form of a board, a set of sectors, a piece of paper, a three-dimensional form, or a screen.
Table 1, provided in
The region 102 is pre-divided into a plurality of sectors 104. The sectors are stationary, and do not move during play of the game. Each sector has an area or volume within the region bounded by pre-defined boundaries. Each sector is distinct and does not overlap with any other sector. Each sector can be any real or imagined bounded shape or form—such as a polygon or other geometric shape, a polyhedron or other three-dimensional form, or a geographical area—within the region. For example, each sector 104 is a triangle. For purposes of the strategy games described herein, a sector cannot be further divided. Every part of the region 102 is defined as being part of a sector, and, collectively, the sectors cover the entire area or volume of the region 102.
Each sector 104 contains a set of elements, each set of elements including one or more elements. The set of elements in a sector is a complete list of the elements contained in the sector, and each element of the set of elements has a type and quantity. The quantity of an element may be any amount, and is preferably greater than zero. In some examples, the type of each element is voters that favor a particular political party. In other examples, the type of each element may be a resource (i.e., something useful), a hazard (i.e., something harmful), or scrap (i.e., something neither useful nor harmful). Each type of element may have the same value to each player. Alternatively, each element may have a value that is player-specific. That is, the same element may be a resource (i.e., something useful) to one player but a hazard (i.e., something harmful) or scrap (i.e., something neither useful nor harmful) to another player.
Referring to
The strategy game 100 shown in
The strategy game 100 shown in
Games of the present technology may be played by at least one player. Some embodiments are designed to be played by a single player, while other embodiments are designed to be played by a plurality of players, such as at least two players. The term “player” as used herein can mean one individual, or a team of individuals. In many games of the present technology, two or more players take alternating turns, and each player must make one move per turn. In some embodiments of the present technology there is only one player who faces an individual challenge. In some games of the present technology, two or more players independently consider the exact same challenge, each using a separate copy of identical game components. In such examples, each player may take turns independently of the other players, using their own set of the game components. Turns may or may not be under time constraints. The player who does the best job of attaining the pre-defined goal wins the game.
Generally, in order to play a game of the present technology, one or more players are provided with a region that has been divided into non-overlapping sectors that (i) may not be further divided, (ii) do not overlap, and (iii) together cover the area or volume of the large region. At the outset, the one or more players are informed of each sector's precise shape, location, and set of elements, and such information may be depicted graphically. In most examples, there are no districts yet created within the region at the start of play, and each sector is initially considered to be unassigned.
During play, each player makes one move per turn by assigning a sector to a district, according to a set of rules defining types of moves that can be made by the at least one player and restrictions governing how districts can be formed from the plurality of sectors, in pursuit of combining the plurality of sectors into a given number of districts in a manner that seeks to achieve a pre-defined goal based on an aggregation of the elements within each district. Examples of pre-defined goals include, but are not limited to: (a) maximizing a portion of the given number of districts that contain at least a certain level of at least one of the elements of the set of elements; (b) minimizing a portion of the given number of districts that contain at least a certain level of at least one of the elements of the set of elements; and (c) maximizing a number of points earned by at least one of the players, where the number of points earned by the at least one player depends on the aggregation of the elements within each district. In examples of strategy games of the present technology that include at least two political parties, such as strategy game 100 of
Rules of strategy games of the present technology define types of moves that can be made by the at least one player and restrictions governing how districts can be formed from the plurality of sectors. For example, the rules may provide at least two general categories of moves that a player can make:
In some examples, the rules include additional categories of moves that are permitted. For example, three additional categories of moves could be:
As another example, rules regarding how districts may be formed may include:
In many games of the present technology, play concludes when there are no more permitted moves, or when every sector within the region has been assigned to a district. If there are two or more players, the winning player is the player that does the best job of achieving the pre-defined goal. If there is one player, the player wins if the pre-defined goal is achieved, and otherwise loses.
Computer games incorporating the concept may be played in at least two modes. In mode 1, a computer plays the role of (i.e., makes the decisions for) one or more players. In this mode, the computer makes use of sophisticated artificial intelligence techniques that are programmed into it ahead of time by a team of expert computer scientists. In mode 2, the computer provides visualization, data storage, and communication services to facilitate play but does not participate as a decision maker during play.
Strategy games of present technology include hundreds of recreational and non-recreational games and puzzles. A taxonomy for the exemplary strategy games is provided below, and is based on a nine-part code. This code summarizes the main aspects of a given example of the present technology. Each part of the code contains one or more capital letters or integers and is separated from the other parts of the code by forward slashes. The generic code for an example of the current technology is as follows:
Part1/Part2/Part3/Part4/Part5/Part6/Part7/Part8/Part9
Part 1 of the code is either the letter “A” or “D.” It is “A” if the example is analog in nature; it is “D” if the example is digital in nature.
Part 2 is either “Z” or “G.” It is “Z” if the example is a single-player puzzle (e.g., an individual challenge like a Sudoku puzzle). It is “G” if the example is a multi-player game.
Part 3 refers to the shape of each sector and the number of sectors. It consists of a letter followed by an integer with no interceding punctuation. It begins with “S” if each sector is a square; “T” if each sector is an equilateral triangle; “H” if each sector is a regular hexagon; “C” if each sector is a complex, two-dimensional shape such as the shape of a real-world county or country; and “0” if each sector is another shape (e.g., a three-dimensional form). The integer Y that follows the letter indicates how many sectors are in the game. If the letter is (S, T, H), the number of sectors is (36*Y, 54*Y, 37*Y) respectively or slightly less than this. If the letter is “C” or “0,” the value Y gives the exact number of sectors.
Part 4 is either “P” or “N.” It is “P” if the example focuses on politics. It is “N” if the example does not focus on politics.
Part 5 is an integer that gives the number of element types that are found within the sectors. Its value often ranges from 2-6.
Part 6 is either the letter “V” or an integer. If it is the letter “V,” the number of districts to be formed is variable and is unknown at the start. Otherwise, the number of districts to be formed is known at the start and equals the value in this part of the code.
Part 7 specifies the game paradigm. It only applies to multi-player games, thus it exists only if part 2 of the code is “G.” Part 7 is “U” if the game involves alternating, turn-based play. It is “I” if the game involves simultaneous independent play in which each player takes turns independently of the others. Games with simultaneous independent play can have any number of players, whereas games with alternating, turn-based play typically have no more than 6 players.
Parts 8 and 9 only apply to multi-player games with alternating, turn-based play; these parts of the code exist only if part 2 of the code is “G” and part 7 of the code is “U.”
Part 8 indicates the number of players in the game. It is expressed as a range—with two integers separated by a hyphen—if different numbers of players can play the game. It is a single integer if the game is designed for a specific number of players (e.g. for two players only).
Part 9 contains one or more of the letters “E,” “X,” “R,” “B,” and “F.” These five letters respectively refer to five categories of allowed moves—“Establish,” “Expand,” “Reassign,” “Break up,” and “Freeze”—which are briefly described in a previous paragraph. This part of the code contains the letters that correspond to the categories of moves that are allowed in the game.
The description of each example provided below begins with a discussion of its taxonomic code. This code gives the reader a quick understanding of the example's main aspects. One or more parts of a code may contain the question mark symbol “?” if those aspects are unspecified.
Several non-limiting examples of strategy games of the present technology are provided below. While the examples use numbers, letters, and generic shapes to distinguish between different districts, element types, and sectors, it should be noted that other methods of distinction could be used. For example, colors or specialized graphics could be used.
Strategy games of the Example 1 have a taxonomic code A/Z/S1/P/2/9. They are analog, single-player puzzles with square sectors and a political focus in which two types of elements are present in the sectors and nine districts are formed. A nearly unlimited number of possible instances of this kind of puzzle can be created, one of which is illustrated in
Two aspects distinguish this kind of puzzle from most other types of logic puzzles. First, there may be multiple solutions to a given puzzle; a unique solution is not guaranteed. Second, the “partial solution” concept does not apply. In other words, if one partially finishes a puzzle, there is no guarantee that the partial solution will give rise to a complete solution. A logical “guess and check” approach is recommended for solving this kind of puzzle.
In strategy game 200, the player is given a map of a square shaped region 202 that has been divided into 36 square shaped sectors 204, which are arranged in six rows and six columns. The player is tasked with dividing the region 200 into a given number of political districts—i.e. to draw lines that define the boundaries of the districts—in order to achieve the stated objective.
Two types of elements—two political parties—occupy the region. One element is the Red Party and the other element is the Blue Party. Each sector 204 may represent a community, and each community has a number which is the community's voter margin. A black number in a white circle means that there are more Red Party supporters than Blue Party supporters in the community (see the key in Table 1). In such a case, the community favors the Red Party. A white number in a black circle means that there are more Blue Party supporters than Red Party supporters in the community. In such a case, the community favors the Blue Party. The number itself is the margin (in thousands of voters) by which the community supports one party over the other. For example, sector 206 has a black 3 in a white circle, which may mean that there are 3000 more Red Party supporters than Blue Party supporters in that community. In this case, the community's voter margin is “+3 Red.” In sector 208, there is a white 4 in a black circle, which may mean that there are 4000 more Blue Party supporters than Red Party supporters in that community. The voter margin in sector 208 is “+4 Blue.” In Sector 210, there is a zero, which means that the community equally supports the two parties. In such a case, the community's voter margin is 0.
In each puzzle of Example 1, the player is asked to divide the region into 9 political districts. In other words, the player is asked to draw lines that define the boundaries of 9 political districts.
The rules for forming political districts are as follows.
When a district is formed, the player must pay attention to its voter margin (i.e. margin). A district's voter margin indicates which party has more voters in the district. A district's voter margin depends on the voter margins of the sectors in the district. It equals the difference between the sum of the black and white numbers in the district. The voter margin favors the Red Party if the sum of the black numbers in white circles exceeds the sum of the white numbers in black circles; it favors the Blue Party if the opposite is true; and it is zero if the sum of the black numbers equals the sum of the white numbers in a district.
For example, if a district has four sectors with voter margins “+4 Red,” “+6 Blue,” “0,” and “+7 Red,” then the district's voter margin is “+5 Red” (=4+7+0−6). In other words, there are 5000 more Red Party supporters than Blue Party supporters in the district. The party with more voters in a district is said to control the district. Neither party controls a district—a district is tied—if the district's voter margin is 0.
In each puzzle, the player may be asked to pursue one of three goals:
Goals A, B, and C relate to the voter margins of the districts that are formed. In exact terms, Goal A is to, first and foremost, maximize the number of districts controlled by the Red Party and, secondarily, maximize the margin by which the Red Party controls its least safe district. Goal B is to do the same except to the benefit of the Blue Party. Goal C is to (i) equalize the number of districts controlled by each party, (ii) equalize the margin by which each party controls its least safe district, and (iii) maximize the number of tied districts that have a voter margin of 0. A party's least safe district is the district in which it has the smallest majority.
Each puzzle may have an easy version and a hard version. The easy version asks the player to pursue the goal at hand—A, B, or C—to a modest extent. The hard version asks the player to pursue the same goal to the maximum possible extent.
There are different solutions to the strategy game 200 for each goal A, B, and C. For example,
Strategy games of the Example 2 have a taxonomic code A/Z/T1/P/2/9. They are analog, single-player puzzles with triangular sectors and a political focus in which two types of elements are present in the sectors and nine districts are formed.
In some examples of this kind of puzzle, the region may be a hexagonal region that has 54 triangular sectors (i.e. communities). Each community may have the same population. An example of such a region is region 102 in
In the example shown in
In each puzzle of Example 2, the player is asked to divide the region into 9 political districts. In other words, the player is asked to draw lines that define the boundaries of 9 political districts.
The rules for forming political districts are as follows.
When a district is formed, the player must pay attention to its voter margin. Just as in Example 1, a district's voter margin equals the difference between the sum of the black and white numbers in the district. The voter margin favors the Red Party if the sum of the black numbers in white circles exceeds the sum of the white numbers in black circles; it favors the Blue Party if the opposite is true; and it is zero if the sum of the black numbers equals the sum of the white numbers in a district.
In each puzzle, the player may be asked to pursue one of three goals:
Each puzzle may have an easy version and a hard version. The easy version asks the player to pursue the goal at hand—A, B, or C—to a modest extent. The hard version asks the player to pursue the same goal to the maximum possible extent.
There are different solutions to the strategy game 100 for each goal A, B, and C. For example,
Strategy games of the Example 3 have a taxonomic code A/G/S1/P/2/9/U/2/EXR. They are analog, multi-player games with square sectors and a political focus in which two types of elements—namely two political parties—are present and nine districts are formed. The game proceeds according to alternating, turn-based play; there are two players; and moves in categories “E,” “X,” and “R” are allowed.
In strategy game 500, each sector represents a community. The region represents a state and has an American-style, two-party political system in which one person is elected to represent each political district. At the start, the districts have not been formed and the players know the location and political composition of each sector (i.e., which party its citizens favor and by how much). During the first phase of the game, players build the political districts by assigning sectors to political districts one sector at a time, in alternating turns. They may also reassign sectors from large districts to adjacent smaller districts in order to better equalize the district sizes. During the optional second phase of the game, the voter margin in each district is converted into a numerical likelihood of each party winning the district, and an election is simulated, which may be done by rolling dice. The winner is the player whose party controls more districts than his/her opponent. A tie is possible if players skip phase 2 of the game.
In the final position, shown in
The table below shows the final result of this game. The Blue Party wins this example game by a score of 5 districts to 3 districts (with one tied district):
In strategy games of this example, the region 502 is laid out and includes boundaries for the sectors. To allow multiple varied games to be played, however, the elements of the sectors are not pre-printed on the region. Instead, sector tiles 610 (
Players may decide (yes or no) if a symmetric game will be played and (yes or no) if phase 2 of the game will be played. The “no-no” option is recommended for beginners. The “yes-no” option is a game of pure skill, whereas the “no-yes” option maximizes the role of luck in the game. Players then decide who plays Red, who plays Blue, and who takes the first turn.
The game components may be laid out at the start of the game is shown in
Each sector tile 610 has a set of elements marked at least once thereon that show the element type (Red Party or Blue Party being favored) and the quantity (voter margin). A black number in a white circle, such as first element set 506 in
As discussed above, in this example there are a total of thirty six sector tiles 610. The seventeen sector tiles favoring the Red party are identical with respect to their voter margins to the seventeen sector tiles favoring the Blue party, and two sectors have a voter margin of 0. Hence, the overall voter margin in the state is 0; the same number of voters support each party statewide.
In this example, the thirty six sector tiles 610—which remain in their initial positions as sectors 504 once placed for the game—are used as building blocks to form nine political districts that will cover the region 502. The nine districts are identified by color: Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, and Gray. Initially, no sector 504 belongs to any district. During the game, players use colored markers to assign communities to political districts. Each community eventually belongs to exactly one political district. Since nine districts will be created from thirty six sectors 504, at the end of the game the size of the average district—the number of sectors it has—will be four. However, the rules may permit variation in the size of a district, so some districts may be smaller or larger than others.
The voter margin of a district depends on the voter margins of the sectors 504 that comprise it. The voter margin of a district equals the difference between the sum of the black and white numbers in the district. The voter margin favors the Red Party if the sum of the black numbers in white circles exceeds the sum of the white numbers in black circles; it favors the Blue Party if the opposite is true; and it is zero if the sum of the black numbers equals the sum of the white numbers in a district. A district's voter margin indicates which party has more voters in the district. For example, if a district has four communities with voter margins “+4 Red,” “+6 Blue,” “0,” and “+7 Red,” then the district's voter margin is “+5 Red” (=4+7+0−6). In other words, there are 5000 more Red Party supporters than Blue Party supporters in the district. The party with more voters in a district is said to control the district. Neither party controls a district—a district is tied—if the district's voter margin is 0.
During the game, the current voter margin of each district is indicated by the position and orientation of its scoring token 606 on the scoreboard 604. In particular, each district's scoring token 606 must always be placed so that (1) the value in the square it occupies plus (2) the number on the side of the scoring token that faces up equals the district's current voter margin. A scoring token may be cubed shaped, and may have its faces marked in the following manner: unmarked, 10, 20, 30, 40, 50. The side of a scoring token 606 that should be face up depends upon the range of the voter margin, and may correspond to unmarked: 0-12, 10:13-22, 20:23-32, 30:33-42, 40:43-52, 50:53-62. The scoring token 606 for a district may be placed on a square within its row 608 when a district's voter margin favors the (Blue, Red) Party respectively. For example, consider a moment in the game when three communities with voter margins “+5 Blue,” “+1 Red,” and “+4 Blue” have been assigned to the Green District. In this case, the Green District's voter margin is “+8 Blue” (=5+4−1), so the green scoring token should be placed on square “Green District Voter Margin=+8 Blue” with its unmarked side facing up. If a community with voter margin “+9 Blue” were added to this district, its new voter margin would be “+17 Blue” (=9+5+4−1), and the green scoring token would be moved to square “Green District Voter Margin=+7 Blue” with its “10” side facing up. Alternatively, if a community with voter margin “+9 Red” were added to this district, its new voter margin would be “+1 Red” (=9+1−5−4), and the green scoring token would be moved to square “+1 Red” with its unmarked side facing up.
Play may include the following three phases, although the second phase is optional.
The first phase is the main phase of the game. During this phase, players may take turns assigning sectors 504 to political districts, one sector at a time, until every sector belongs to a political district. The assignment of a sector 504 to a political district is accomplished by placing a home base marker 600 or expansion marker 602 on a vacant sector 504. Players may also reassign sectors from large districts to adjacent smaller districts to better equalize the district sizes. This is done by changing the color of the marker on a sector 504. At the end of this phase, there will be 9 non-overlapping political districts—Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, and Gray—that cover the region 502.
Each district evolves in the same way. Initially, it is formless. At some point, it is established when its home base marker 600 is placed on a vacant sector 504. (A vacant sector is a sector with no marker on it.) It is then expanded whenever one of its expansion markers 602 is placed on a vacant sector 504 that is adjacent to a sector 504 that already belongs to the district.
The process of building political districts is relatively unrestricted. There is no general requirement for the sequence in which, or locations where, districts are constructed. Once begun, the construction of a district may be temporarily halted while players take turns establishing, expanding, and/or resizing other districts. There is no district size requirement. However, the rules encourage the creation of districts having four sectors 504.
Importantly, all marker subsets 620 (consisting of the home base marker and expansion markers for a given color) and all sectors 504 are available to all players. No player “owns” any marker subset or sector 504. As long as the rules below are followed, any player may contribute to building any district during any turn. No matter which player established a district, any other player may expand the district or reassign a sector 504 from that district to another district.
Note: During play, the voter margins of all sectors 504 are visible to both players. In
Players may take alternating turns beginning with the starting player. During a player's turn, he/she (A) makes one move and then (B) records the move on the scoreboard. Forfeiting a turn (i.e. passing on a turn) is not allowed.
All moves must be of type 1, 1A, 2, 2A, 3, or 3A (described below). Moves of type 1 and 1A establish a new district. These moves are in category E. Moves of type 2 and 2A expand an existing (i.e. already established) district. These moves are in category X. Moves of type 3 and 3A resize two adjacent districts. These moves are in category R. “A” means “alternate move.”
Play is divided into three stages. In stage 1, only moves of type 1, 2, and 3 are allowed. Play enters stage 2 if (i) no moves of type 1 exist and (ii) fewer than nine districts have been established. In stage 2, only moves of type 1A, 2, and 3 are allowed. Play enters stage 3 immediately after the 9th district is established. In stage 3, the next move must be of type 2 or 3A if a move of type 2 exists. Otherwise, the next move must be of type 2A or 3A. In many games, stage 2 is skipped and play proceeds directly from stage 1 to stage 3. Play concludes when no legal moves exist.
The six types of legal moves are as follows.
Two sectors are adjacent—and connected—if and only if they share a common edge. For example, the two-sector area shown in (i) in
In this game, every political district must be connected at all times. That is, at all times and for any two sectors that belong to a given district (say District X), there must be a path within District X—a sequence of adjacent sectors that all belong to District X—connecting those two sectors.
A set of connected, vacant sectors is captured if it is (i) surrounded by a single district or (ii) surrounded by the edge of the board on one side and a single district on the other side. In
A move of type 2 which captures exactly one sector is allowed if the district's new size—including the sector on which the marker is placed and the sector that is captured—is no greater than four sectors. All other moves that capture sectors are forbidden. For example, if sector 1 is vacant, it is permissible to add sector 2 to a district consisting of sectors 7-8. In this case, sector 1 is captured and the new district consists of sectors 1-2 and 7-8. However, if sectors 5-6 are vacant, adding sector 12 to a district consisting of sectors 4 and 10-11 is not allowed.
A sector that is captured during a legal move of type 2 is immediately assigned to the district that has captured it. An expansion marker is immediately placed on this sector, and the scoreboard is updated appropriately.
A district is trapped if (i) it (and the open spaces beside it) is either surrounded by a single district or is surrounded by the edge of the board on one side and a single district on the other side and (ii) its size (in sectors) plus the sizes of the open spaces beside it is less than four.
In
A move of type 2 which traps a district is forbidden. For example, if the Gray District consists of sector 1 and the Green District consists of sectors 7-8, then an expansion of the Green District to sector 2 is not allowed. Also, placing a blue expansion marker on sector 15 to achieve the position in
Phase 1 ends when no legal moves exist. When this happens, exactly one marker will occupy each sector, and the state will be partitioned into nine political districts that average four sectors each.
This optional phase of the game accounts for the surprises that can happen in real-world elections. Sometimes the candidate whose party has the majority of voters in a district is defeated by his/her opponent. This may happen if a candidate lacks charisma, public speaking skills, good looks, or other personal qualities or if the candidate takes unpopular stands on issues such as education, health care, the economy, infrastructure, foreign affairs, the environment, etc. In this phase of the game, the voter margin in each district is converted into a numerical likelihood of each party winning the district, and an election in each district is simulated by rolling dice 618.
Each district is considered one at a time beginning with the Brown District.
First, using the table below, the voter margin for the party with more voters in the district is converted into a numerical likelihood of that party winning an election in the district. For example, a “+8 Red” voter margin in the Yellow District converts to a 97% chance for the Red Party to win an election in the Yellow District.
Second, a random number from 1-100 is produced by simultaneously rolling the two 10-sided dice. The result shown on the black (white) die is the value of the tens (ones) digit of the random number. For example, if the black (white) die shows 7 (1), the result is 71. If the black (white) die shows 0 (8), the result is 8. The only exception to the above rule is that a roll of “zero-zero” gives the result of 100.
Third, the random number is compared to the winning percentage (e.g. 97 for the above case). If the random number is less than or equal to the winning percentage, the party with more voters in the district wins the district election. If the random number is greater than the winning percentage, the party with fewer voters in the district wins the district election. In the above example, the Red Party wins the Yellow District election if the random number is from 1-97, and the Blue Party wins the Yellow District election if the random number is from 98-100. If both parties have a 50% chance of winning a district, the Blue Party wins if the random number is from 1-50 and the Red Party wins if the random number is from 51-100. If a party has a 100% chance of winning a district, it automatically wins that district without a dice roll. After the winner of an election is identified, the scoring token that matches the district color is placed on the “Blue Wins” or “Red Wins” square in that district's portion of the scoreboard 604.
The above procedure is repeated for each of the nine districts.
In the game's final phase, the overall winner is identified.
If phase 2 is played, the winner is the player whose party wins five or more district elections. If phase 2 is not played, players identify the party that controls each district, i.e. the party with more voters in each district. This is done by looking at the positions of the scoring tokens on the scoreboard. The winner is the player whose party controls more districts than his/her opponent. If the players control an equal number of districts, the result is a tie.
During stage 1 of play, only moves of type 1, 2, and 3 are allowed. After 15 turns, assume the position in
The available moves of type 1 in this position are as follows:
The available moves of type 2 in this position are as follows:
No moves of type 3 are available in this position.
Notes:
Eight moves later in the game, after a total of 23 moves, the new position is shown in
The available moves of type 1 in this position are as follows:
The available moves of type 2 in this position are as follows:
No moves of type 3 are available in this position.
Notes:
Four moves later, after a total of 27 moves, the new position is shown in
The available moves of type 1A in this position are as follows:
The available moves of type 2 in this position are as follows:
No moves of type 3 are available in this position.
Notes:
Two moves later, after a total of 29 moves, the new position is shown in
The available moves of type 2 in this position are as follows:
The available moves of type 3A in this position are as follows:
Notes:
Three moves later, after a total of 32 moves, the new position is shown in
The available moves of type 2A in this position are as follows:
The available moves of type 3A in this position are as follows:
Notes:
Two moves later. After a total of 34 moves, the new position is shown in
The available moves of type 2 in this position are as follows:
The available moves of type 3A in this position are as follows:
One move later, after a total of 35 moves, the new position is shown in
The available moves of type 2A in this position are as follows:
The available moves of type 3A in this position are as follows:
One move later, after a total of 36 moves, the new position is shown in
The available moves in this position are as follows:
One move later, after a total of 37 moves, the new position is shown in
The final position at the end of phase 1 is shown in
The final district voter margins are shown on the scoreboard (see table below).
If phase 2 is not played, the game immediately ends, and the Blue Party wins by a score of 5 districts to 3 districts (with one tied district).
If phase 2 is played, dice 618 are rolled to determine the winning party in each district. In the game at hand, the final voter margin of the (Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, Gray) District is (0, +7 Blue, +3 Blue, +8 Red, +5 Red, +2 Blue, +9 Red, +1 Blue, +9 Blue). Using a preceding table, these margins translate to winning likelihoods of (50%, 95%, 77%, 97%, 90%, 69%, 99%, 60%, 99%) for the parties with the majority of voters in these districts respectively. Note that each party has a 50% chance of winning the Brown District, and no party automatically wins a district with 100% probability.
Dice 618 are then thrown to determine the election results. The results are summarized in the table below. Despite being at a disadvantage going into the election, the Red Party “gets lucky” and wins the elections in five out of nine districts. The Red Party wins the game by a score of 5 districts to 4 districts.
A starting region with a large connected portion of high-numbered sectors favoring the Blue Party but no large connected portion of high-numbered sectors favoring the Red Party is biased in favor of the Red Party. In such a setting, the player representing the Red Party will more easily be able to concentrate or “pack” the voting power of the opposing party into a small number of districts than the player representing the Blue Party. Thus, the Red Party is more likely to win the game.
The purpose of a symmetric game is to remove bias from the initial sector arrangement and give each party—Red and Blue—a fair chance of winning the game. This is particularly important in a tournament setting.
A symmetric game has three additional rules compared to a regular game. Rule 1 creates a symmetric initial sector arrangement, and rules 2 and 3 minimize the possibility of a symmetric position during play. The three rules are as follows.
This example of the present technology a game of pure skill if (A) players decide who plays first prior to the start of the game, (B) a symmetric game is played, and (C) phase 2 of the game is skipped. This form of the game, like international chess and the Japanese game go, is highly suited to tournament play. Unlike chess and go, the initial board position in this game is always different, so every game has a unique opening.
This game is suited to handicap play. If the players' skill levels differ, the playing field can be leveled by changing the voter margin of one or more sectors. For example, if the stronger player represents the Blue Party, the players may agree, before any sector tiles 610 are placed, to change the voter margin of the first “+8 Blue” sector tile that is placed from “+8 Blue” to 0. Alternatively, the weaker player may be allowed to make more than 50% of the moves—for example 5 of every 9 moves.
In one variation of strategy game 500, seven districts are formed instead of nine districts. In this variation, only seven marker subsets 620, representing seven district colors, are used, and the average size of a district at the end of the game is about five sectors 504. At the end of the game, the region will be divided into seven political districts. This variation of the game may be played according to the same rules above except that the requirements for moves of type 1, 1A, and 2 are slightly different as described below:
Strategy games of Example 4 have a taxonomic code A/G/S2/P/2/12/U/2/EXR. They are analog, multi-player games with 72 square sectors and a political focus in which two types of elements—namely two political parties—are present and twelve districts are formed. The games proceed according to alternating, turn-based play; there are two players; and moves in categories “E,” “X,” and “R” are allowed.
One example is strategy game 700 as shown in
Strategy game 700 is very similar to strategy game 500 described in Example 3. The main differences are as follows. First, in strategy game 700, there are 72 sectors—exactly twice as many sectors of each kind as in strategy game 500. Second, at the start of strategy game 700 the sector tiles 716 are randomly placed in a 9×8 rectangular arrangement within region 702 to form sectors 704. Third, twelve districts—the average size of which at the end of the game may be six sectors 704—will be formed. A set of markers 726 containing a total of 12 marker subsets 710 may be used, each marker subset 710 representing a district and consisting of one home base marker 712 and eighteen expansion markers 714. As shown there are nine rectangular home base markers (one of each for Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, and Gray) and 162 square expansion markers (18 of each of the same colors as the rectangular home base markers), and then three additional home base markers that are diamond shaped (one each for the Brown2, Red2, and Orange2 Districts) and three additional sets of expansion markers that are triangular in shape (18 of each of the same colors as the diamond shaped home base markers). Each district is distinguishable by the color and/or shape of the markers used to form it. Fourth, the precise rules for making moves of types 1, 1A, 2, and 2A are slightly different in this game to encourage most districts to have a size of six sectors 704 at the end of the strategy game 700.
In the illustrated example, strategy game 700 includes two scoreboards, a first scoreboard 706 and a second scoreboard 708. Each scoreboard has a plurality of rows 722 and looks like
As illustrated,
The table below shows the final result of this game. The Blue Party wins this example game by a score of 6 districts to 5 districts (with one tied district).
Strategy game 700 may have the same three phases as, and may be played in a manner that is nearly identical to, Example 3.
During a player's turn, he/she (A) makes one move and (B) records the move on the appropriate scoreboard 706 or 708. All moves must be of type 1, 1A, 2, 2A, 3, or 3A below. About 80 moves—40 by each player—are made in a game. The game ends when no legal moves exist. The winner is the player whose party controls more districts than his/her opponent.
Play may be divided into three stages. In stage 1, only moves of type 1, 2, and 3 are allowed. Play enters stage 2 if (i) no moves of type 1 exist and (ii) fewer than 12 districts have been established. In stage 2, only moves of type 1A, 2, and 3 are allowed. Play enters stage 3 immediately after the 12th district is established. In stage 3, the next move must be of type 2 or 3A if a move of type 2 exists. Otherwise, the next move must be of type 2A or 3A. In many games, stage 2 is skipped and play proceeds directly from stage 1 to stage 3. Play concludes when no legal moves exist. Forfeiting a turn (i.e. passing on a turn) is not allowed.
The rules may provide six types of legal moves, such as those listed below. The terms “captured” and “connected” have the same meaning as described in Example 3. Moves of type 3 and 3A are identical to Example 3.
Phase 1 ends when no legal moves exist. When this happens, exactly one marker will occupy each sector, and the state will be partitioned into 12 political districts that average 6 sectors each.
This phase of the game is nearly identical to Example 3 except that a different table (shown below) is used to convert a district's voter margin into the probability that the party with more voters in the district wins an election in the district.
This phase of the game is exactly the same as in Example 3. If phase 2 is played, the winner is the player whose party wins seven or more district elections. If each party wins six district elections, the result is a tie. If phase 2 is not played, players look at the scoring tokens on the scoreboard to identify the party that controls each district, i.e. the party with more voters in each district. The winner is the player whose party controls more districts than his/her opponent. If the players control an equal number of districts, the result is a tie.
Game Alternative #1: Form 18 Districts (with an Average Size of 4 Sectors)
One variation of strategy game 700 includes the formation of 18 districts—instead of 12—in the same 9×8 region 702. In this variation, exactly twice as many home base, expansion, and scoring tokens are used compared to Example 3. The size of the average district at the end of this variation of the game is 4 sectors 704. This variation may be played according to the same rules above except that the requirements for moves of type 1, 1A, and 2 may be slightly different as described below:
As with strategy game 500, strategy game 700 can be played as a symmetric game. The purpose of a symmetric game is to remove bias from the initial sector arrangement and give each party—Red and Blue—a fair chance of winning the game.
When strategy game 700 is played as a symmetric game, there are two additional rules that may be used compared to a regular game. Rule 1 creates a symmetric initial sector arrangement, and rule 2 reduces the possibility of a symmetric position during play. The two rules are as follows.
Strategy games of Example 5 have a taxonomic code A/G/S3/P/2/15/U/2/EXR. They are analog, multi-player games with 90 square sectors and a political focus in which two types of elements—namely two political parties—are present and 15 districts are formed. These games proceed according to alternating, turn-based play; there are two players; and moves in categories “E,” “X,” and “R” are allowed.
Strategy games of Example 5 are is very similar to the games described in Examples 3-4. In one example, two players—Red and Blue—vie for political control of a 9×10 rectangular state by competitively creating 15 political districts (whose average size is 6) out of 90 square communities. The game can be played with any 90 sectors in which the sets of red and blue sectors are identical—for example 5 each of sector tiles “+1 Red” to “+8 Red” and “+1 Blue” to “+8 Blue” (80 sector tiles); 3 each of sector tiles “+9 Red” and “+9 Blue” (six sector tiles); and 4 sector tiles with a voter margin of 0. Players take alternating turns beginning with the starting player. During a player's turn, he/she (A) makes one move and then (B) records the move on two scoreboards. All moves must be of type 1, 1A, 2, 2A, 3, or 3A below. About 100 moves—50 by each player—are made in a game. The game ends when no legal moves exist. The winner is the player whose party controls more districts than his/her opponent.
Play is divided into three stages. In stage 1, only moves of type 1, 2, and 3 are allowed. Play enters stage 2 if (i) no moves of type 1 exist and (ii) fewer than 15 districts have been established. In stage 2, only moves of type 1A, 2, and 3 are allowed. Play enters stage 3 immediately after the 15th district is established. In stage 3, the next move must be of type 2 or 3A if a move of type 2 exists. Otherwise, the next move must be of type 2A or 3A. Play concludes when no legal moves exist.
The six types of moves allowed in the rules for games of this type are summarized below.
Strategy games of Example 6 are larger versions of the games described in Examples 3-5. These games have a taxonomic code A/G/S4/P/2/15/U/2/EXR. They are analog, multi-player games with 121 square sectors and a political focus in which two types of elements—namely two political parties—are present and 15 districts are formed. These games proceed according to alternating, turn-based play; there are two players; and moves in categories “E,” “X,” and “R” are allowed.
In at least one example, two players—Red and Blue—vie for political control of an 11×11 square region by competitively creating 15 political districts (whose average size is just above 8) out of 121 square sectors, each of which represents a community. The game can be played with any 121 sectors in which the sets of red and blue sectors are identical—for example 7 each of sector tiles “+1 Red” to “+8 Red” and “+1 Blue” to “+8 Blue” (112 sector tiles); 3 each of sector tiles “+9 Red” and “+9 Blue” (six sector tiles); and 3 sector tiles with a voter margin of 0. Players take alternating turns beginning with the starting player. During a player's turn, he/she (A) makes one move and then (B) records the move on two scoreboards. All moves must be of type 1, 1A, 2, 2A, 3, or 3A below. About 140 moves—70 by each player—are made in a game. The game ends when no legal moves exist. The winner is the player whose party controls more districts than his/her opponent.
Play is divided into three stages. In stage 1, only moves of type 1, 2, and 3 are allowed. Play enters stage 2 if (i) no moves of type 1 exist and (ii) fewer than 15 districts have been established. In stage 2, only moves of type 1A, 2, and 3 are allowed. Play enters stage 3 immediately after the 15th district is established. In stage 3, the next move must be of type 2 or 3A if a move of type 2 exists. Otherwise, the next move must be of type 2A or 3A. Play concludes when no legal moves exist.
The six types of legal moves are summarized below.
Strategy games of Example 7 are larger versions of the games described in Examples 3-6. They have a taxonomic code A/G/S5/P/2/21/U/2/EXR. They are analog, multi-player games with 169 square sectors and a political focus in which two types of elements—namely two political parties—are present and 21 districts are formed. Games of this type proceed according to alternating, turn-based play; there are two players; and moves in categories “E,” “X,” and “R” are allowed.
In one example, two players—Red and Blue—vie for political control of a 13×13 region by competitively creating 21 political districts (whose average size is just above 8) out of 169 square sectors. The game can be played with any 169 sectors in which the sets of red and blue sectors are identical—for example 10 each of sector tiles “+1 Red” to “+8 Red” and “+1 Blue” to “+8 Blue” (160 sector tiles); 3 each of sector tiles “+9 Red” and “+9 Blue” (six sector tiles); and 3 sector tiles with a voter margin of 0. Players take turns beginning with the starting player. During a player's turn, he/she (A) makes one move and then (B) records the move on three scoreboards. All moves must be of type 1, 1A, 2, 2A, 3, or 3A below. About 180 moves—90 by each player—are made in a game. The game ends when no legal moves exist. The winner is the player whose party controls more districts than his/her opponent.
Play is divided into three stages. In stage 1, only moves of type 1, 2, and 3 are allowed. Play enters stage 2 if (i) no moves of type 1 exist and (ii) fewer than 21 districts have been established. In stage 2, only moves of type 1A, 2, and 3 are allowed. Play enters stage 3 immediately after the 21st district is established. In stage 3, the next move must be of type 2 or 3A if a move of type 2 exists. Otherwise, the next move must be of type 2A or 3A. Play concludes when no legal moves exist.
The six types of legal moves are summarized below.
This is a larger version of the games described in Examples 3-7. This game has taxonomic code A/G/S6/P/2/21/U/2/EXR. It is an analog, multi-player game with 210 square sectors and a political focus in which two types of elements—namely two political parties—are present and 21 districts are formed. The game proceeds according to alternating, turn-based play; there are two players; and moves in categories “E,” “X,” and “R” are allowed.
This game is very similar to Examples 3-7. In this game, two players—Red and Blue—vie for political control of a 15×14 rectangular state by competitively creating 21 political districts (whose average size is 10) out of 210 square communities. The game can be played with any 210 sectors in which the sets of red and blue sectors are identical—for example 12 each of sector tiles “+1 Red” to “+8 Red” and “+1 Blue” to “+8 Blue” (192 sector tiles); 6 each of sector tiles “+9 Red” and “+9 Blue” (12 sector tiles); and 6 sector tiles with a voter margin of 0. Players take turns beginning with the starting player. During a player's turn, he/she (A) makes one move and then (B) records the move on three scoreboards. All moves must be of type 1, 1A, 2, 2A, 3, or 3A below. About 240 moves—120 by each player—are made in a game. The game ends when no legal moves exist. The winner is the player whose party controls more districts than his/her opponent.
Play is divided into three stages. In stage 1, only moves of type 1, 2, and 3 are allowed. Play enters stage 2 if (i) no moves of type 1 exist and (ii) fewer than 21 districts have been established. In stage 2, only moves of type 1A, 2, and 3 are allowed. Play enters stage 3 immediately after the 21st district is established. In stage 3, the next move must be of type 2 or 3A if a move of type 2 exists. Otherwise, the next move must be of type 2A or 3A. Play concludes when no legal moves exist.
The six types of legal moves are summarized below.
Strategy games of Example 9 have a taxonomic code A/G/S1/P/2/9/I. They are analog, multi-player games with 36 square sectors and a political focus in which two types of elements—namely two political parties—are present and 9 districts are formed. The game paradigm is simultaneous independent play, in which each player makes one move per turn in sequential turns, independently of the other players. Any number of players—two or more—may play.
In one example, illustrated in
The players decide (yes or no) if phase 2 of the game will be played, and they agree upon a time limit for each phase of the game. The “yes” option with a 10-minute time limit is recommended. (Such a game lasts about 50 minutes.)
One player may be selected as the leader. All players except the leader may organize their 36 sector tiles 820 into 19 face-up piles—one pile for each number+color combination—so that specific sector numbers and colors can be quickly located. The leader may spread their sector tiles 820 out face down, mix them, and organize them face down into a single deck. The leader may then draw the 36 sector tiles 820 from the deck one at a time and place them face up with one on each sector placeholder 806 of the region 802 to form sectors 804. Each time a sector is drawn, the leader may announce its (a) position in the sequence, (b) color, and (c) number—for example “Sector 1: Red 4,” “Sector 2: Blue 2,” “Sector 3: Zero,” etc.—so that every other player may find the same sector tile 820 from his/her game set and place it in the same location in his/her region 802. When this process ends, each player has a copy of the leader's sector arrangement in his/her region 802. One possible arrangement of sectors 804 in the region 802 is identical to that shown in
Each player may use his/her scoreboard 810 and scoring tokens 812 to keep track of the voter margins of the districts that he/she creates during play.
At the end of each phase of the game, the leader or other players may visit each player's playing area to ensure that the positions and orientations of his/her scoring tokens properly show the voter margins of the districts that he/she has created. Each district's scoring token 812 should be placed so that (1) the value in the square it occupies on the scoreboard plus (2) the number on the side of the scoring token that faces up equals the district's voter margin. When the scoring tokens 812 are cubes—having sides that are unmarked and marked 10, 20, 30, 40, 50 respectively—the (unmarked, 10, 20, 30, 40, 50) side of a scoring token should face up if the voter margin of its district is in the range (0-12, 13-22, 23-32, 33-42, 43-52, 53-62) respectively. The scoring token 812 is placed on the scoreboard to reflect the voter margin in each district. For example, if Player 1's Green District contains sectors with voter margins “+6 Blue,” “+1 Red,” “+9 Blue,” and “+7 Blue,” then the Green District's voter margin is “+21 Blue” (=6+9+7−1) and Player 1's green scoring token should be placed on the square “Green District Voter Margin=+11 Blue” with its “10” side facing up. If Player 2's Gray District contains sectors with voter margins “+3 Blue,” “+1 Red,” “+9 Red,” and “+5 Blue,” then the Gray District's voter margin is “+2 Red” (=1+9−3−5) and Player 2's gray scoring token should be placed on the square “Gray District Voter Margin=+2 Red” with its unmarked side facing up.
Play may consist of the following two phases. The second phase is optional.
The timer 814 is set to the time limit agreed upon by the players, if time limits are being used. Play begins when the timer starts or the players agree to begin.
During the first phase of the game, each player independently uses his/her colored markers to form 9 political districts—Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, and Gray—on his/her 36 sectors 804. Each district is formed by placing four markers, one per turn, that match the district color on four adjacent sectors. Each player's main goal in this phase is to create 9 political districts—i.e. a district plan—in which the Red Party controls as many districts as possible. A party controls a district if it has the majority—strictly more than half—of the voters in a district. Each player's secondary goal is to make the “voter margin in the district that the Red Party controls by the least amount” as high as possible.
The rules may require that each player's district plan must satisfy the following two requirements:
Each player is free to use his/her scoreboard, game board, and markers as desired. It is recommended that each player (a) use the expansion markers 808 beside his/her 36 sectors 804 and scoreboard to create and evaluate potential district plans and (b) use the duplicate region 824 on his/her game board 818 to store the best district plan that he/she has found. At the end of this phase, each player's final district plan must be displayed by a set of 36 markers (four per color) that are placed either on his/her region 802 or on the duplicate region 824 on his/her game board 818.
When play concludes, the final district plan made by each player is scored. The scoring of each player's final district plan is done by (a) computing the district voter margins, (b) placing scoring tokens appropriately on the scoreboard, and (c) computing the following values:
If a player's district plan violates requirement A or B above, he/she receives scores of 0 and 1 for items 1 and 2 respectively.
One example of the result of phase 1 of a two-player version of game 800 is shown in
At this point, Player 1's scoreboard should show that the voter margin of his/her (Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, Gray) District is (+3 Red, +2 Red, +10 Red, +18 Blue, +16 Blue, +12 Red, +5 Blue, +4 Red, +8 Red). The Red Party controls 6 districts, and the lowest voter margin in those six districts is “+2 Red.” Player 2's scoreboard should show that the voter margin of his/her (Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, Gray) District is (+2 Red, +7 Blue, +7 Red, +2 Red, +3 Red, +3 Red, +5 Red, +19 Blue, +4 Red). The Red Party controls 7 districts, and the lowest voter margin in those seven districts is “+2 Red.” Overall, Player 2 has done better in this phase of the game because his/her district plan gives the Red Party control of more districts than Player 1's district plan. Each player uses two red markers to mark his/her scores for phase 1 in the first two rows 828 of his/her game board 818 as shown below.
Phase 2: Build Political Districts that Maximize the Blue Party's Advantage
At the end of phase 1, players remove all markers from their scoreboards and game boards, except the two red markers used to mark their final scores for phase 1 on their game boards. The timer, if used, is then set to the time limit agreed upon by the players. Play of phase 2 is then started.
During phase 2, play proceeds exactly as in phase 1 except that now each player's (i) main goal is to create a district plan in which the Blue Party controls as many districts as possible and (ii) secondary goal is to make the “voter margin in the district that the Blue Party controls by the least” as high as possible. The rules for making districts are just as in phase 1.
When the timer goes off, all players cease their activities, disengage from their playing areas, and assemble as a group in the middle of the room. Working as a team, the players together (a) compute the district voter margins, (b) place scoring tokens appropriately on the scoreboard, and (c) compute the following for each player's final district plan:
If a player's district plan violates requirement A or B above, he/she receives scores of 0 and 1 for items 3 and 4 respectively.
The above quantities are remembered by placing two blue markers on the appropriate squares in rows 3 and 4 of each player's game board.
One example of the result of phase 2 of the two-player version of game 800 is shown in
At this point, Player 1's scoreboard should show that the voter margin of his/her (Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, Gray) District is (+5 Blue, +1 Blue, +10 Blue, +2 Blue, +1 Blue, +6 Red, +5 Blue, +4 Blue, +22 Red). The Blue Party controls 7 districts, and the lowest voter margin in those seven districts is “+1 Blue.” Player 2's scoreboard should show that the voter margin of his/her (Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, Gray) District is (“+10 Red,” “+15 Blue,” “+6 Blue,” “+3 Blue,” “+25 Red,” “+3 Blue,” 0, “+3 Blue,” “+5 Blue”). The Blue Party controls 6 districts, and the lowest voter margin in those six districts is “+3 Blue.” Overall, Player 1 has done better in this phase of the game because his/her district plan gives the Blue Party control of more districts than Player 2's district plan. Each player uses two blue markers to mark his/her scores for phase 2 in rows 3-4 of his/her game board 818 as shown below.
The winner of the game is identified, by process of elimination, by looking at the markers in the rows 828 of each player's game board 818. These markers may show the scores for the following items:
If phase 2 is not played, the winner is identified as follows. First, every player whose district plan does not tie for having the highest score for item 1 above is eliminated. Next, among the remaining players, every player whose district plan does not tie for having the highest score for item 2 above is eliminated. Any player who is not eliminated wins the game.
If phase 2 is played, the winner is identified as follows. First, every player whose district plan does not tie for having the highest sum of scores for items 1+3 is eliminated. Next, among the remaining players, every player whose district plan does not tie for having the highest sum of scores for items 2+4 is eliminated. Any player not eliminated wins the game. In the preceding illustrative example, the players' scores for items 1+3 have the same sum, so the sum of the scores for items 2+4 is the tiebreaker. The sum of Player 1's scores for items 2+4 is 3. The sum of Player 2's scores for items 2+4 is 5, so Player 2 wins.
Strategy games of Example 10 are similar to those of Example 9 and have a taxonomic code A/G/S1/P/2/9/I. They are analog, multi-player games with 36 square sectors and a political focus in which two types of elements—namely two political parties—are present and 9 districts are formed. These games proceed according to simultaneous independent play, so any number of players may play.
In one example, the game is set up exactly as described in Example 9, but the pre-defined goal of each player is to create the most balanced set of political districts. In particular, each player's pre-defined goal may be to create a district plan that (i) equalizes the number of districts controlled by each party, (ii) equalizes the margin by which each party controls its least safe district, and (iii) maximizes the number of tied districts that have a voter margin of 0. Item (i) has priority over (ii), and (ii) has priority over (iii). Each player's district plan must satisfy requirements A-B (see description of Example 9).
At the end of play, each player's final district plan must be displayed and scored. Scoring may include (a) computing the district voter margins, (b) placing scoring tokens appropriately on the scoreboard, and (c) computing the following for each player's final district plan:
The above items may be remembered by placing two red, two blue, and three gray markers in appropriate places on the rows of each player's game board 828.
The winner is identified by process of elimination. First, every player whose district plan violates one of the requirements A-B (see description of Example 9) is eliminated. Second, among the remaining players, every player whose district plan does not tie for having the lowest score for item 5 above is eliminated. Next, among the remaining players, every player whose district plan does not tie for having the lowest score for item 6 above is eliminated. Finally, among the remaining players, every player whose district plan does not tie for having the highest score for item 7 above is eliminated. Any player who is not eliminated wins the game. If all players' district plans violate one of the requirements A-B (see description of Example 9), all players lose.
Player 1's scoreboard should show that the voter margin of his/her (Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, Gray) District is (+19 Blue, +5 Blue, 0, +2 Blue, +12 Red, +8 Blue, +4 Red, +7 Red, +11 Red). In player 1's district plan, the Red and Blue Party each control 4 districts; the lowest voter margin in the districts controlled by the Red Party is 4; the lowest voter margin in the districts controlled by the Blue Party is 2; and one district is tied. Player 2's scoreboard should show that the voter margin of his/her (Brown, Red, Orange, Yellow, Green, Blue, Purple, Pink, Gray) District is (0, +3 Blue, +15 Blue, +14 Blue, +3 Red, +25 Red, +15 Blue, +9 Red, +10 Red). In player 2's district plan, the Red and Blue Party each control 4 districts; the lowest voter margin in the districts controlled by the Red Party is 3; the lowest voter margin in the districts controlled by the Blue Party is 3; and one district is tied.
The scoring of items 1-7 above takes place in rows 1-7 of the game board and is summarized in the table below. Both players' district plans satisfy requirements A-B, and the players have the same score for item 5, so the score for item 6 is the tiebreaker. Player 2 has a lower score for item 6, so Player 2 wins the game.
Strategy games of Example 11 are generally for 2-4 players and are similar to games of Example 3. They have taxonomic code A/G/S1/P/4/9/U/2-4/EXR. They are analog, multi-player games with 36 square sectors in which four types of elements—such as four political parties—are present and nine districts are formed. These games proceed according to alternating, turn-based play, and moves in categories “E,” “X,” and “R” are allowed.
The table below shows the total number of districts that each party wins outright (W) and ties for winning (T). It also shows the total points earned by each party assuming that 3 points are earned when a party wins a district outright and 1 point is earned when a party ties for winning a district. In the final tally, the Blue Party wins this game with 8 points (2 outright wins+2 ties).
The components for this game are similar to those used in Example 3. They are as follows.
The setup is similar that in Example 3. The main difference is that, in this game, four scoring tokens—one for each color—are stacked on the position “No. Voters for Each Party=0” in each district's portion of the scoreboard.
Play may include of the following three phases. The second phase is optional.
This phase proceeds exactly as in Example 3. The only difference is that the turns alternate among up to four players instead of two.
Phase 2 is somewhat different in a four-party version of strategy game 900 as compared to strategy game 500 in Example 3. In this phase of the game, the political status in each district is converted into a numerical likelihood of each party winning the district, and an election in each district is simulated by rolling dice. In particular, the three-step procedure below (A-B-C) is performed for each district beginning with the Brown District.
(A) The table below is used to convert the district's political status into a probability of each party winning an election in the district. This is done by (1) ranking the parties according to voter support in the district (i.e. deciding which party is in 1st, 2nd, 3rd and 4th place); (2) computing the difference in voter support between the parties; and (3) finding the appropriate row in the table below. If two or more parties have the same voter tally, their ranking is inconsequential but the players must still (arbitrarily) rank them. For example, if the political status of the Green district in a four-player game is (3 Red Voters, 3 Orange Voters, 4 Green Voters, 2 Blue Voters), then the Green (Red, Orange, Blue) Party is in 1st (2nd, 31d, 4th) place, the difference between 1st and 2nd place is 1 voter, the difference between 2nd and 3rd place is 0 voters, and the difference between 3rd and 4th place is 1 voter. In this case, the Green (Red, Orange, Blue) Party has a 60% (20%, 20%, 0%) chance of winning an election in the Green District. In a 2-3 player game, parties not represented by an active player are not ignored. These parties can still win districts and earn points at the end of the game. However, such parties are not allowed to win the game. Only a party represented by an active player may win the game.
(B) Next, a random number from 1-100 is produced by simultaneously rolling the two 10-sided dice. The value on the black (white) die is the tens (ones) digit of the random number. For example, if the black (white) die shows 7 (1), the result is 71. If the black (white) die shows 0 (8), the result is 8. The only exception to the above rule is that a roll of “zero-zero” gives the result of 100. In the unlikely event that the result is 100 and three parties are tied for having the most voters in the district, the dice should be re-rolled until a result below 100 is obtained.
(C) The winner of the district's election is then determined by comparing the random number to the parties' winning percentages in the district. If the random number is less than or equal to the winning percentage of the 1st place party, the 1st place party wins the district election. Otherwise, if the random number is less than or equal to the sum of the winning percentages of the 1st and 2nd place parties, the 2nd place party wins the district election. Otherwise, if the random number is less than or equal to the sum of the winning percentages of the 1st, 2nd, and 3rd place parties, the 3rd place party wins the district election. If the 1st, 2nd, and 3rd place parties have the same number of voters in a district and the result is 100, the dice are re-rolled until a value below 100 is obtained. Otherwise, if the random number is greater than the sum of the winning percentages of the 1st, 2nd and 3rd place parties, the 4th place party wins the district election. In the example described in step A, the Green Party wins the Green District election if the random number is any value from 1-60; the Red Party wins the Green District election if the random number is any value from 61-80; and the Orange Party wins the Green District election if the random number is any value from 81-100. All scoring tokens are then removed from that district's portion of the scoreboard, and a single marker matching the color of the party that wins the election that is placed on the “Red Wins,” “Orange Wins,” “Green Wins,” or “Blue Wins” square in that district's portion of the scoreboard.
The above procedure is repeated for each of the nine political districts.
We consider the (4-player) game whose final board position is shown in
Dice 618 are then rolled to determine the winners of the seven districts in which there is
not an automatic winner. The election results are summarized in the table below. In the table, the dice rolls are shown in the “Dice Roll” column, and “R,” “O,” “G,” and “B” refer to the Red, Orange, Green, and Blue Party respectively. The overall result is that the Blue (Red, Green) Party wins the elections in three (two, four) districts.
In the game's final phase, the overall winner is identified. If phase 2 was played, the winner is the player whose party wins the most district elections. If more than one party ties for winning the most district elections, these parties together win the game and the result is a tie. For example, the Green Party wins the game shown in the table above.
If phase 2 was not played, the scoreboard is used to identify the party that controls each district. Two or more parties jointly control a district if they tie for having the greatest number of voters in the district. If there is no tie, the party with the greatest number of voters in the district solely controls the district. Each party receives three points for each district that it solely controls and one point for each district that it jointly controls. The player whose party has more points than any other player's party is the winner. If parties represented by two or more players tie for having the most points, those players jointly win. A party not represented by a player may not win the game. Refer to section “Game Summary” above to see who wins the game whose final position is shown in
This example encompasses several games that are larger versions of the game described in Example 11. These larger versions are played with more sectors than Example 11 but are otherwise very similar to Example 11. The relationship of these games to Example 11 is analogous to the relationship of Examples 4-8 to Example 3.
The taxonomic codes for five possible games included in this example are listed below. All games are analog, multi-player games with square sectors and a political focus in which four types of elements—namely four political parties—are present, the game proceeds according to alternating turn-based play, there are 2-4 players, and moves in categories “E,” “X,” and “R” are allowed.
The aforementioned five games get progressively larger with 72, 90, 121, 169, and 210 sectors respectively.
The move types allowed in the aforementioned five games are identical to the move types described in Examples 4-8 respectively.
The main difference between these games and Examples 4-8 is that up to four players can play these games.
Play of any of the above games proceeds in a manner similar to Example 11. During the first phase of the game, players build the political districts by assigning sectors to political districts one sector at a time. They may also reassign sectors from large districts to adjacent smaller districts in order to better equalize the district sizes. During the (optional) second phase of the game, the political status of each district is converted into a numerical likelihood of each party winning the district, and an election is simulated by rolling dice. In the game's final phase, the parties that control the districts are identified, and 3 (1) points are awarded to a party that has sole (joint) control of a district. The player whose party has more points than any other player's party is the winner. If parties represented by two or more players tie for having the most points, those players jointly win.
Strategy games of Example 13 are a combination of the simultaneous independent play undertaken in Example 9 and the four-party environment considered in Example 11. Such games have taxonomic code A/G/S1/P/4/9/I. They are analog, multi-player games with 36 square sectors and a political focus in which four types of elements—namely four political parties—are present and nine districts are formed. The games proceed according to simultaneous independent play, so any number of players may participate.
In one example, the game components are highly similar to the components used in Examples 9 and 11 and shown in
The setup is very similar to that in Example 9, but with each player laying out a region 902 having sectors 904. Overall, each player creates a copy of the exact same 6×6 sector arrangement in his/her playing region 902 and organizes piles of markers within his/her playing area to prepare for what follows.
Play may include four phases, each having a different pre-defined goal, as listed below. Phases 2-4 are optional.
Each phase proceeds like a phase described in Example 9. Players use their markers, scoreboard, and game board as desired to try to achieve the pre-defined goal. The main goal in each phase is to create 9 political districts—i.e. a district plan—in which the concerned party controls as many districts as possible. A party controls a district if it has strictly more voters in a district than any other party. Each player's secondary goal is to maximize the total amount—summed over the districts controlled by the concerned party—by which the concerned party leads its closest adversary in the districts that it controls.
At the end of each phase of the game, each player tracks his/her score with respect to the pre-defined goals above by placing markers on squares in relevant rows 828 of his/her game board 818. Penalties are assessed if a player's district plan violates requirement A or B (see Example 9).
The winner of the game is identified, by process of elimination, by looking at the markers on the rows 828 of each player's game board 818. These markers show the scores for up to eight items:
If all phases are played, the winner is identified as follows. First, every player whose district plan does not tie for having the highest sum of scores for items 1+3+5+7 is eliminated. Next, among the remaining players, every player whose district plan does not tie for having the highest sum of scores for items 2+4+6+8 is eliminated. Any player not eliminated wins the game.
Strategy games of Example 14 combine the (optionally) time-limited, simultaneous independent play undertaken in Example 10 and the four-party environment considered in Example 11. Such games have a taxonomic code A/G/S1/P/4/9/I. They are analog, multi-player games with 36 square sectors and a political focus in which four types of elements—namely four political parties—are present and nine districts are formed. These games proceed according to simultaneous independent play, so any number of players may participate.
In one example, the game components are nearly identical to those in Example 13. The only difference is that a few additional markers are needed to track the final score on the game board.
Setup (about 10 Minutes)
The setup is very similar to that in Example 9. Overall, each player creates a copy of the exact same 6×6 sector arrangement in his/her region.
The scoreboard and scoring tokens may be the same as in Example 11.
Play proceeds just as in Example 10. As long as time has not expired, players may use their markers, scoreboard, and game board as desired to try to achieve the desired goal. Each player's main goal is to create 9 political districts—i.e. a district plan—in which all four parties control the same number of districts. A party controls a district if it has strictly more voters in a district than any other party. Each player's secondary goal is to equalize the total amount—summed over the districts controlled by each party—by which each party leads its closest adversary in the districts that it controls. Each player's tertiary goal is to maximize the number of districts in which all four parties have the same number of voters.
At the end of play (e.g., when time expires), each player tracks his/her score with respect to the 3 goals above by placing markers on squares in the left part of his/her game board. Penalties are assessed if a player's district plan violates requirement A or B (see Example 9).
The winner of the game is identified by looking at the markers on the left side of each player's game board. These markers show the scores for eleven items:
The winner is identified by process of elimination. First, every player whose district plan violates one of the requirements A-B (see description of Example 9) is eliminated. Second, among the remaining players, every player whose district plan does not tie for having the lowest score for item 9 above is eliminated. Next, among the remaining players, every player whose district plan does not tie for having the lowest score for item 10 above is eliminated. Finally, among the remaining players, every player whose district plan does not tie for having the highest score for item 11 above is eliminated. Any player who is not eliminated wins the game. If all players' district plans violate one of the requirements A-B (see description of Example 9), all players lose.
Strategy games of Example 15 have taxonomic code A/G/T1/P/2/9/U/2/EXR and are triangular-sector games similar to the square-sector games of Example 3. They are analog, multi-player games with 54 sectors in the shape of an equilateral triangle. They have two types of elements—such as two political parties—and nine districts are formed. The games proceed according to alternating, turn-based play; there are two players; and moves in categories “E,” “X,” and “R” are allowed.
During the first phase of strategy game 1600, players build the political districts by assigning sectors to political districts one sector at a time. They may also reassign sectors from large districts to adjacent smaller districts in order to better equalize the district sizes. During the optional second phase of strategy game 1600, the political margin in each district is converted into a numerical likelihood of each party winning the district, and an election is simulated by rolling dice 1622. The winner is the player whose party controls more districts than his/her opponent. A tie is possible if players skip phase 2 of the game.
The game components are as follows:
The game has three phases and plays in a manner similar to Example 3.
In phase 1, players take alternating turns beginning with the starting player. During a player's turn, he/she (A) makes one move and then (B) records the move on two scoreboards. Forfeiting a turn (i.e. passing on a turn) is not allowed. All moves must be of type 1, 1A, 2, 2A, 3, or 3A below. About 60 moves—30 by each player—are made in phase 1.
Phase 1 is divided into three stages. In stage 1, only moves of type 1, 2, and 3 are allowed. Play enters stage 2 if (i) no moves of type 1 exist and (ii) fewer than 9 districts have been established. In stage 2, only moves of type 1A, 2, and 3 are allowed. Play enters stage 3 immediately after the 9th district is established. In stage 3, the next move must be of type 2 or 3A if a move of type 2 exists. Otherwise, the next move must be of type 2A or 3A. Phase 1 concludes when no legal moves exist.
The rules may allow districts to be formed by the following six types of moves. The meanings of the phrases “connectedness,” “captured tile,” and “trapped district” are analogous to those in Example 3.
Phases 2-3 are played almost exactly as in Example 3.
A symmetric version of this game may be played if players are concerned about bias in the initial sector arrangement. The purpose of a symmetric game is to remove bias from the initial sector arrangement and thereby give each party—Red and Blue—a fair chance of winning the game.
A symmetric game has three additional rules compared to a non-symmetric game. Rule 1 guarantees a symmetrical initial sector arrangement, whereas rules 2 and 3 minimize the possibility of a symmetric board position during play. The three rules are as follows.
Strategy games of Example 16 are larger versions of the games described in Example 15. These games have taxonomic code A/G/T2/P/2/12/U/2/EXR. They are analog, multi-player games with 96 triangular sectors in which 12 districts are formed. Such games proceed according to alternating, turn-based play; there are two players; and moves in categories “E,” “X,” and “R” are allowed.
In one example, a strategy game may be very similar to Example 15. In such a game, two players—Red and Blue—vie for political control of the hexagonal region by competitively creating 12 political districts (whose average size is 8) out of 96 triangular sectors. The sectors may be pre-established on the region, or formed by placing one sector tile on each of 96 sector placeholders on the region. The game can be played with any 96 triangular sectors, though it is preferred that the sets of sectors favoring red and blue be identical—for example 6 each of sector tiles “+2 Red” to “+9 Red” and “+2 Blue” to “+9 Blue” (96 sectors total). Players take alternating turns beginning with the starting player. During a player's turn, he/she (A) makes one move and then (B) records the move on two scoreboards. All moves must be of type 1, 1A, 2, 2A, 3, or 3A below. About 100 moves-50 by each player—are made in a game. The game ends when no legal moves exist. The winner is the player whose party controls more districts than his/her opponent.
Play is divided into three stages. In stage 1, only moves of type 1, 2, and 3 are allowed. Play enters stage 2 if (i) no moves of type 1 exist and (ii) fewer than 12 districts have been established. In stage 2, only moves of type 1A, 2, and 3 are allowed. Play enters stage 3 immediately after the 12th district is established. In stage 3, the next move must be of type 2 or 3A if a move of type 2 exists. Otherwise, the next move must be of type 2A or 3A. Play concludes when no legal moves exist.
The six types of legal moves are summarized below.
Strategy games of Example 17 are even larger versions of the games described in Example 15. These games have taxonomic code A/G/T3/P/2/15/U/2/EXR. They are analog, multi-player games with 150 triangular sectors, and may have a political focus in which two types of elements—namely two political parties—are present and 15 districts are formed. These games may proceed according to alternating, turn-based play; may have two players; and moves in categories “E,” “X,” and “R” may be allowed.
During the first phase of the game, players build the political districts by assigning sectors 1004 to political districts, one sector per turn. They may also reassign sectors during a turn from large districts to small districts in order to better equalize the district sizes. During the (optional) second phase of the game, the political margin in each district is converted into a numerical likelihood of each party winning the district, and an election is simulated by rolling dice. The winner is the player whose party controls more districts than his/her opponent. A tie is possible if players skip the game's second phase.
The game components may be as follows:
The play is very similar to Example 15 but is more challenging owing to the many sectors. Play may include the following three phases. The second phase is optional.
During the first phase of strategy game 1000, players take turns assigning/reassigning sectors to political districts, one sector at a time, until every sector belongs to a political district. The assignment of a sector 1004 to a political district is accomplished by placing a home base marker 1010 or expansion marker 1012 on a sector tile. Reassignment of a sector 1004 from a district to another district may be accomplished by changing the color of the marker on a sector tile. At the end of the first phase, 15 political districts (e.g., Brown, Red, Orange, Yellow, Light Green, Dark Green, Light Blue, Dark Blue, Purple, Pink, Light Gray, Dark Gray, Black, White, and Gold) will be formed on the region 1002.
Each district evolves in the same general way. Initially, it is formless. At some point, it is established when its home base marker 1010 is placed on a vacant sector 1004. It is then expanded whenever one of its expansion markers 1012 is placed on a vacant sector that is adjacent to a sector that already belongs to the district. Later, it may be resized so its size is more similar to neighboring districts by reassignment of sectors 1004.
The process of building political districts is relatively unrestricted. There is no general requirement for the sequence in which, or locations where, districts are constructed. Once begun, the construction of a district may be temporarily halted while players take turns establishing, expanding, and/or resizing other districts. There may not be any district size requirement. However, the rules may encourage or require the creation of districts of size 10 (meaning that each district is formed from 10 sectors 1004).
Importantly, all marker subsets and all sectors are available to all players. No player “owns” any marker subset or sector. As long as the rules below are followed, any player may contribute to building any district during any turn. No matter which player established a district, any other player may expand the district or reassign a sector from that district to another district.
Players take turns beginning with the starting player. During a player's turn, he/she (A) makes one move and then (B) records the move on the scoreboard.
All moves must be of type 1, 1A, 2, 2A, 3, or 3A. Moves of type 1 and 1A establish a new district. Moves of type 2 and 2A expand an existing (i.e. already established) district. Moves of type 3 and 3A resize two adjacent districts. “A” means “alternate move.”
Play is divided into three stages. In stage 1, only moves of type 1, 2, and 3 are allowed. Play enters stage 2 if (i) no moves of type 1 exist and (ii) fewer than 15 districts have been established. In stage 2, only moves of type 1A, 2, and 3 are allowed. Play enters stage 3 immediately after the 15th district is established. In stage 3, the next move must be of type 2 or 3A if a move of type 2 exists. Otherwise, the next move must be of type 2A or 3A. In many games, stage 2 is skipped and play proceeds directly from stage 1 to stage 3. Phase 1 concludes when no legal moves exist.
The six types of legal moves are as follows. The meanings of the phrases “connectedness,” “captured tile,” and “trapped district” are analogous to those in Example 3.
The second phase of strategy game 1000 is optional. In the second phase, the political margin in each district is converted into a numerical likelihood of each party winning the district, and an election in each district is simulated by rolling the dice.
Each district shown in the final position (e.g., in
Next, a random number from 1-100 is produced by simultaneously rolling the two 10-sided dice. See the section “Phase 2: Run an election” in Example 3 for more details.
The random number is then compared to the winning percentage. If the random number is less than or equal to the winning percentage, the party with more voters in the district wins the district election. If the random number is greater than the winning percentage, the party with fewer voters in the district wins the district election. If both parties have a 50% chance of winning, the Blue Party wins if the random number is 1-50 and the Red Party wins if the random number is 51-100. After the winner of an election is identified, the unused scoring token that matches the district color is placed on the “Blue Wins” or “Red Wins” square in that district's row in the scoreboard.
The above procedure is repeated for each of the 15 political districts.
In the game's final phase, the winner is identified. If phase 2 is played, the winner is the player whose party wins eight or more district elections.
If phase 2 is not played, the scoreboard is used to identify the party that controls each district, i.e. the party with more voters in each district. The winner is the player whose party controls more districts than his/her opponent. If the two players control an equal number of districts, the result is a tie.
As with many previous examples, strategy game 1000 can be played as a symmetric game. The purpose of a symmetric game is to remove bias from the initial sector arrangement and give each party—Red and Blue—a fair chance of winning the game.
A symmetric game for strategy game 1000 has three additional rules compared to a regular game. Rule 1 creates a symmetric initial sector arrangement, and rules 2 and 3 reduce the possibility of a symmetric position during play. The three rules are as follows.
Strategy games of this Example 18 are even larger versions of the games with hexagonal regions and triangular sectors described in Examples 15-17. These games have taxonomic code A/G/T4/P/2/27/U/2/EXR. They are analog, multi-player games with 216 triangular sectors, and may have a political focus in which two types of elements—namely two political parties—are present and 27 districts are formed. These games proceed according to alternating, turn-based play; there are two players; and moves in categories “E,” “X,” and “R” are allowed.
In one example, two players—Red and Blue—vie for political control of giant hexagonal region by competitively creating 27 political districts (whose average size is 8) out of 216 triangular communities. The game can be played with any 216 triangular sectors in which the sets of red and blue sectors are identical—for example 12 each of sector tiles “+2 Red” to “+9 Red” and “+2 Blue” to “+9 Blue” (192 sector tiles); 8 each of sector tiles “+1 Red” and “+1 Blue” (16 sector tiles); and 8 sector tiles with a voter margin of 0. Players take turns beginning with the starting player. During a player's turn, he/she (A) makes one move and then (B) records the move on three scoreboards. All moves must be of type 1, 1A, 2, 2A, 3, or 3A below. About 240 moves—120 by each player—are made in a game. The game ends when no legal moves exist. The winner is the player whose party controls more districts than his/her opponent.
Play is divided into three stages. In stage 1, only moves of type 1, 2, and 3 are allowed. Play enters stage 2 if (i) no moves of type 1 exist and (ii) fewer than 27 districts have been established. In stage 2, only moves of type 1A, 2, and 3 are allowed. Play enters stage 3 immediately after the 27th district is established. In stage 3, the next move must be of type 2 or 3A if a move of type 2 exists. Otherwise, the next move must be of type 2A or 3A. Play concludes when no legal moves exist.
This game has the same sector shape and same final average district size as Example 16. Thus, the details for the six types of legal moves—listed below—are identical to Example 16.
Strategy games of Example 19 combine the simultaneous independent play undertaken in Example 9 and the triangular sectors used in Example 15. These games have taxonomic code A/G/T1/P/2/9/I. They are analog, multi-player games with 54 triangular sectors and a political focus in which two types of elements (e.g., two political parties) are present and nine districts are formed. The game paradigm is simultaneous independent play, so any number of players may participate.
The game components are highly similar to the components used in Example 15, shown in
Play may include of the following two phases. Phase 2 is optional.
Each phase proceeds like a phase described in Example 9. Players use their markers, scoreboard, and game board as desired to try to achieve the desired goal. The main goal in each phase is to create 9 political districts of equal size—i.e. a district plan—in which the concerned party controls as many districts as possible. Each player's secondary goal is to make the “voter margin in the district that the concerned party controls by the least amount” as high as possible. In the district plan, (A) each sector 1604 must be assigned to exactly one district and (B) each district must consist of six connected tiles as in
At the end of each phase of the game, each player tracks his/her score with respect to the two goals above by placing markers on the relevant square of the relevant row 1104 of his/her game board 1100. Penalties are assessed if a player's district plan violates requirement A or B above.
The winner of the game is identified, by process of elimination, by looking at the markers on the left side of each player's game board. These markers show the scores for up to four items:
If phase 2 is not played, the winner is identified as follows. First, every player whose district plan does not tie for having the highest score for item 1 above is eliminated. Next, among the remaining players, every player whose district plan does not tie for having the highest score for item 2 above is eliminated. Any player who is not eliminated wins the game.
If phase 2 is played, the winner is identified as follows. First, every player whose district plan does not tie for having the highest sum of scores for items 1+3 is eliminated. Next, among the remaining players, every player whose district plan does not tie for having the highest sum of scores for items 2+4 is eliminated. Any player not eliminated wins the game.
Strategy games of Example 20 combine the simultaneous independent play undertaken in Example 10 and the triangular sectors used in Example 15. These games have taxonomic code A/G/T1/P/2/9/I. They are analog, multi-player games with 54 triangular sectors and a political focus in which two types of elements—two political parties—are present and 9 districts are formed. The playing paradigm is simultaneous independent play, so any number of players may play.
In one example, the game is set up exactly as described in Example 19, but the pre-defined goal of each player is to create the most balanced set of political districts. The region has an American-style, two-party political system in which one person is elected to represent each political district. At the outset, the districts are formless and the players know the political status of each sector (i.e. which party its citizens favor and by how much). During the game, players simultaneously and independently work on identical copies of the region map to create political districts that equalize the political advantage of the two parties, Red and Blue. The winner is the player who creates the most balanced set of political districts.
The game components, setup, sectors, scoreboard, and scoring methods are identical to Example 19 except that each player uses an additional 3 gray markers to score three additional items on his/her game board.
Play proceeds as in Example 19, except with the pre-defined goal to create a district plan that (i) equalizes the number of districts controlled by each party, (ii) equalizes the margin by which each party controls its least safe district, and (iii) maximizes the number of tied districts that have a voter margin of 0. Item (i) has priority over (ii), and (ii) has priority over (iii). Each player's district plan must satisfy requirements A-B as stated in the section “Playing the game” in the description of Example 19.
At the conclusion of play, the players (a) compute the district voter margins, (b) place scoring tokens appropriately on the scoreboard, and (c) compute the following for each player's final district plan:
The winner is identified by process of elimination. First, every player whose district plan violates one of the requirements A-B (see description of Example 19) is eliminated. Second, among the remaining players, every player whose district plan does not tie for having the lowest score for item 5 above is eliminated. Next, among the remaining players, every player whose district plan does not tie for having the lowest score for item 6 above is eliminated. Finally, among the remaining players, every player whose district plan does not tie for having the highest score for item 7 above is eliminated. Any player who is not eliminated wins the game. If all players' district plans violate one of the requirements A-B (see description of Example 19), all players lose.
Strategy games of Example 21 combine the alternating, turn-based play for more than two players from Example 11 with the triangular sectors of Example 15. The taxonomic code for these games is A/G/T1/P/3/9/U/2-3/EXR. They are analog, multi-player games with 54 triangular sectors and a political focus in which three types of elements—three political parties—are present and nine districts are formed. The games proceed according to alternating, turn-based play; there are 2-3 players; and moves in categories “E,” “X,” and “R” are allowed.
The mechanics of this game are slightly different than in previous examples. In this game, the same types of moves—1, 1A, 2, 2A, 3, and 3A—are allowed during play, but there are additional restrictions regarding their timing. In particular, all moves of type 1, 1A, 2, and 2A must be completed during phase 1 of this game—i.e. the board must be completely full—before the first move of type 3 or 3A is allowed to be made in phase 2. The district size threshold in moves of type 2 also differs from earlier examples.
Play may include the following four phases. Phases 2-3 are optional.
This phase proceeds almost exactly like phase 1 in Example 15. The main differences are that in this game (a) turns alternate among up to three players instead of two players and (b) only moves of type 1, 1A, 2, and 2A are available.
All moves made in phase 1 must be of type 1, 1A, 2, or 2A below.
Play during the first phase is divided into four stages. During stage 1, only moves of type 1 and 2 are allowed. Play enters stage 2 if (i) no more moves of type 1 exist and (ii) fewer than nine districts have been established. During stage 2, only moves of type 1A and 2 are allowed. Play enters stage 3 immediately after the 9th district is established. During stage 3, only moves of type 2 are allowed. Play enters stage 4 if no more moves of type 2 exist. During stage 4, only moves of type 2A are allowed. This phase of the game ends when every sector has been assigned to a political district.
The four types of legal moves during this phase of the game are as follows. Note that the “5-sector district size restriction” and “no tile capturing restriction” in moves of type 2 is slightly different than in Example 15 and other previous examples.
This optional phase of the game is motivated by the need to keep the populations of real-world political districts nearly equal. In this phase of the game, players take turns modifying the sector-to-district assignments in order to better equalize the district sizes (which are a proxy for the district populations).
Players take turns beginning with the player to the left of the player who took the final turn during phase 1. During a player's turn, he/she makes a move by changing the district to which one sector is assigned. This is done by removing the (home base or expansion) marker that occupies one sector tile and replacing it with an expansion marker of a different color. The net result is that one district loses a sector and one district gains a sector. The other seven districts remain unchanged. The player then updates the scoreboard to reflect the move that has been made.
Every move made during this phase of the game must be of type 3A:
This phase of the game concludes when no more moves of type 3A exist.
Phase 3 in this game is optional and is very similar to phase 2 in Example 11. During this phase of the game, the political status of each district is converted into a numerical likelihood of each party winning the district, and an election is simulated by rolling dice.
In the game's final phase, the overall winner is identified. If phase 3 was played, the winner is the player whose party wins the most district elections. If more than one party ties for winning the most district elections, these parties together win the game and the result is a tie.
If phase 3 was not played, the winner is determined by identifying the party that controls each district, i.e. the party with the most voters in each district. Each party receives 6 points for each district that it solely controls; 3 points for each district that it jointly controls with one other party; and 2 points for each district that it jointly controls with two other parties. The winner is the player whose party has more points than any other player's party. If parties represented by two or more players tie for having the most points, those players jointly win.
A symmetric version of this game may be played if there is a desire to eliminate bias in the initial sector arrangement. A symmetric game has three additional rules compared to a regular game. Rule 1 guarantees an initial sector arrangement that is symmetric, whereas rules 2 and 3 minimize the possibility of a symmetric position during play. The three rules are as follows.
Strategy games of this example encompass several games that are played with more sectors than Example 21 but are otherwise very similar to Example 21.
The taxonomic codes for three possible games included in this example are listed below. All games are analog, multi-player games with triangular sectors and a political focus in which three types of elements—namely three political parties—are present, the game proceeds according to alternating turn-based play, there are 2-3 players, and moves in categories “E,” “X,” and “R” are allowed.
The three games above get progressively larger with 96, 150, and 216 sectors respectively.
Strategy games of Example 23 combine the simultaneous independent play of Example 19 with the three-party environment in Example 21. These games have taxonomic code A/G/T1/P/3/9/I. They are analog, multi-player games with 54 triangular sectors and a political focus in which three types of elements—three political parties—are present and nine districts are formed. The game paradigm is simultaneous independent play, so any number of players may participate.
In one example, the game components are similar to the components used in Example 21. A digital or mechanical timer is needed to play this game. In addition, each player should have a copy of the same game set which contains (a) 54 sectors having the same markings as in Example 21, (b) dozens of markers, (c) a scoreboard, and (d) a game board.
Play may include the following four phases. Phases 2-4 are optional.
Each phase 1-3 proceeds like a phase in Example 19. Phase 4 proceeds as in Example 20.
At the end of each phase of the game, each player tracks his/her score with respect to the goal at hand by placing markers on the appropriate squares on his/her game board.
The winner is the player who does the overall best job of achieving the goals that were pursued during the different phases of the game.
Strategy games of Example 24 are played with hexagonal sectors. Some examples of these games have taxonomic code A/G/H1/P/2/9/U/2/EXR. They are analog, multi-player games with 37 hexagonal sectors and a political focus in which two types of elements—two political parties—are present and nine districts are formed. Taxonomic codes for five additional games included in this example are listed below. All five games are analog, multi-player games with hexagonal sectors and a political focus in which two types of elements—two political parties—are present, the game proceeds according to alternating turn-based play, and there are 2 players.
The five games above get progressively larger with 61, 91, 127, 169, and 217 sectors respectively. The shape of the region in each game is essentially a regular hexagon.
The strategy games of this example proceed according to alternating, turn-based play; there are two players; and moves in categories “E,” “X,” and “R” are allowed.
Rules that can be used for strategy games of this type are generally similar or identical to those of previous examples.
Strategy games of Example 25 have taxonomic code A/G/H1/P/2/9/I. They are analog, multi-player game with 37 hexagonal sectors and a political focus in which two types of elements—two political parties—are present and nine districts are formed. These games proceed according to simultaneous independent play, so any number of players may participate.
Rules that can be used for strategy games of this type are generally similar or identical to those of previous examples with two parties and simultaneous independent play.
Strategy games of Example 26 have taxonomic code A/G/H1/P/6/9/U/2-6/EXR. They are analog, multi-player games with 37 hexagonal sectors and a political focus in which six types of elements—six political parties—are present and nine districts are formed. These games proceed according to alternating, turn-based play; there are 2-6 players; and moves in categories “E,” “X,” and “R” are allowed.
Rules that can be used for strategy games of this type are generally similar or identical to those in Examples 11 and 21.
Strategy games of Example 27 have taxonomic code A/G/H1/P/6/9/I. They are analog, multi-player games with 37 hexagonal sectors and a political focus in which six types of elements—six political parties—are present and nine districts are formed. These games proceed according to simultaneous independent play, and a variety of pre-defined goals can be pursued in each phase of these games.
Rules that can be used for strategy games of this type are generally similar or identical to those in Examples 13-14 and 23.
Strategy games of Example 28 have a real-world focus in which U.S. congressional districts are formed in a real U.S. state, namely Wisconsin. These games have taxonomic code A/G/C82/P/3/8/U/2/EXR. They are analog, multi-player games with 82 complex sectors and a political focus in which three types of elements—population, Red Party supporters, and Blue Party supporters—are considered and eight districts are formed. These games proceed according to alternating, turn-based play; there are 2 players; and moves in categories “E,” “X,” and “R” are allowed.
The components of the game are listed below:
The players decide (yes or no) if phase 2 of the game will be played. The players then decide who plays Red and who plays Blue, and who will take the first turn.
As can be seen in
In this game, the 82 sectors 1404 are used as building blocks to form eight non-overlapping political districts which together exhaust the land area of the state. The eight districts are identified by color: Brown, Red, Orange, Yellow, Green, Blue, Purple, and Gray. Initially, the political districts are formless and no sector belongs to any district. During the course of the game, players use markers 1418 to gradually assign these 82 sectors to political districts. Each sector eventually belongs to exactly one political district.
A close inspection of the sectors 1404 as shown in
Scoring tokens are used to display the (voter) population and political margin of every district on scoreboard (
The scoreboard should be updated after every player takes a turn. For example, consider a moment in the game when exactly three sectors with populations 40, 71, and 48 and voting tendencies “+7 Red,” “+8 Blue,” and “+10 Red” respectively have been assigned to the Green District. In this case, the Green District's population—159—should be indicated by three green scoring tokens placed at positions “Population x100=1,” “Population x10=5,” and “Population x1=9” in the Green District's portion of the scoreboard. The Green District's current political marging—“+9 Red”—equals the difference between the sum of the numbers in the sectors that support the Red Party (e.g. 17) and the sum of the numbers in the sectors that support the Blue Party (e.g. 8) that belong to the district. The political margin favors the Red Party if there are more Red Party than Blue Party supporters in the district; it favors the Blue Party if the opposite is true. A district's political margin is indicated by four scoring tokens that show (1) which party has the majority of voters in the district, (2) the hundreds digit of the political margin, (3) the tens digit of the political margin, and (4) the ones digit of the political margin. In the case above, four green scoring tokens should be placed at the positions “Current Leader=Red,” “Political Margin x100=0,” “Political Margin x10=0,” and “Political Margin x1=9” in the Green District's portion of the scoreboard. If a sector with population 72 and voting tendency “+36 Blue” is added to this district, the district's new population is 231 and its new political margin is “+27 Blue,” so the seven green scoring tokens should immediately be moved to positions “Population x100=2,” “Population x10=3,” “Population x1=1,” “Current Leader=Blue,” “Political Margin x100=0,” “Political Margin x10=2,” and “Political Margin x1=7.”
Play may include the following three phases (the second phase is optional):
This is the main phase of the game. During this phase, players take turns assigning sectors 1404 to political districts, one sector at a time, until every sector 1404 belongs to a political district. The assignment of a sector to a political district is accomplished by placing a home base marker or expansion marker on a sector. Players may also reassign sectors from more populated districts to less populated districts in order to better equalize the district populations. This is done by changing the color of the marker that occupies a sector. At the end of this phase, there will be eight non-overlapping political districts—Brown, Red, Orange, Yellow, Green, Blue, Purple, and Gray—that together cover the state. Also, each district will be connected.
Each district evolves in the same general way. Initially, it is formless. At some point, the district is established when its home base marker is placed on a vacant sector. (A vacant sector is a sector with no marker on it.) The district is then expanded whenever one of its expansion markers is placed on a vacant sector that is adjacent to a sector that already belongs to the district. Later, the district may be adjusted so its population is more similar to neighboring districts.
The overall process of building the political districts is relatively unrestricted. In general, any player may contribute to building any district during any of his/her turns. There is no requirement for the sequence in which, or locations where, districts are constructed. Once begun, the construction of a given district may be temporarily halted while players take turns establishing, expanding, and/or adjusting other districts. There is no district population requirement. However, the rules encourage the creation of districts whose population is close to the average value of 349.
Players take turns beginning with the starting player. During a player's turn, he/she (A) makes one move and then (B) records the move on the scoreboard. Forfeiting a turn is not allowed.
The rules of this game provide that all moves made during this phase of the game must be of type 1, 1A, 2, 2A, 3, or 3A below. Moves of type 1 and 1A establish a new district. Moves of type 2 and 2A expand an existing district. Moves of type 3 and 3A adjust two adjacent districts by transferring a sector from one district to an adjacent district. “A” stands for “alternate move.”
Phase 1 of the game is divided into three stages. During stage 1, only moves of type 1, 2, and 3 are allowed. Play enters stage 2 if (i) no more moves of type 1 exist and (ii) fewer than eight districts have been established. During stage 2, only moves of type 1A, 2, and 3 are allowed. Play enters stage 3 immediately after the 8th district is established. During stage 3, the next move must be of type 2 or 3A if at least one move of type 2 exists. Otherwise, the next move must be of type 2A or 3A. This phase of the game ends when no more legal moves exist.
The six types of legal moves are as follows. Explanations for the asterisked terms are provided at the end of these descriptions.
Two sectors are adjacent—and connected—if and only if they share a common edge. For example, in the map shown in
In this game, every political district must be connected at all times. That is, at all times and for any two sectors that belong to a given district (say District X), there must be a path within District X (i.e. a sequence of adjacent sectors that all belong to District X) connecting those two sectors.
A set of connected, vacant sectors is captured if it is (i) surrounded by the edge of the board on one side and a single district on the other side or (ii) entirely surrounded by a single district. In
A move of type 2 which captures one or more sectors is allowed if the district's initial population before the move is 348 or less and the district's new population—including the sector where the marker is placed and any sectors that are captured—is 398 or less. All other moves that capture sectors are forbidden. For example, in
Any sectors that are captured during a legal move of type 2 are immediately assigned to the district that has captured them. Expansion markers are immediately placed on these sectors.
A district is trapped if (i) it (and the open spaces beside it) is surrounded either by the edge of the board on one side and a single district on the other side or by a single district on all sides and (ii) its population plus the populations of all open spaces beside it is 348 or less. In
A move of type 2 which traps a district is forbidden. For example, if the Brown District consists of sectors 10, 12, 20, 22, 26-27, and 34 and the Orange District consists of sector 11, then an expansion of the Brown District to sector 8 is forbidden.
In this phase, the political margin in each district is converted into a numerical likelihood of each party winning the district, and an election in each district is simulated by rolling the two 10-sided dice.
Each district is considered one at a time. First, using the table below, the political margin for the party with more voters in the district is converted into a numerical likelihood of that party winning an election in the district. For example, a “+4 Blue” political margin in the Yellow District converts to a 66% chance for the Blue Party to win an election in the Yellow District.
Next, a random number from 1-100 is produced by rolling the two 10-sided dice.
The random number is then compared to the winning percentage (e.g. 66 for the above case). If the random number is less than or equal to the winning percentage, the party with more voters in the district wins the district election. If the random number is greater than the winning percentage, the party with fewer voters in the district wins the district election. In the above example, the Blue Party wins the Yellow District election if the random number is from 1-66, and the Red Party wins the Yellow District election if the random number is from 67-100. If both parties have a 50% chance of winning the election, the Blue Party wins if the random number is 1-50 and the Red Party wins if the random number is 51-100. After the winner of an election is identified, the unused scoring token that matches the district color is placed on the “Winner=Blue” or “Winner=Red” square in the appropriate district's portion of the scoreboard.
The above procedure is repeated for each political district.
In the game's final phase, the overall winner is identified.
If phase 2 is played, the winner is the player whose party wins five or more district elections. If each party wins four district elections, the result is a tie.
If phase 2 is not played, the scoreboard is used to identify the party that controls each district, i.e. the party with more voters in each district. The winner is the player whose party controls more districts than his/her opponent. If the two players control an equal number of districts, the result is a tie.
An example of play is now provided, with reference to
During stage 1, only moves of type 1, 2, and 3 are allowed. After 40 moves have been made—six of type 1 and 34 of type 2—assume the board position, with home base markers 1422 and expansion markers 1424, is as shown in
There are 43 possibilities for the next move which must be of type 1, 2, or 3. No legal moves of type 3 exist because no district is confined. Regarding moves of type 1, it is possible to establish a new district in sector 73 or 74. Establishing a new district in another sector is not allowed because either (i) the sector is less than three steps away from a previously placed home base marker or (ii) there is not enough space to grow the new district into a connected district with a population of at least 349. Regarding moves of type 2, it is possible to (a) expand the Brown District to sector 5, 13, 22, 21, 19, 56, 50, 46, 28, 29, 30, or 16; (b) expand the Red District to sector 28, 46, 51, 57, 53, 40, 38, or 37; (c) expand the Orange District to sector 19, 24, 31, 32, 42, 43, 54, 60, 76, 65, 63, or 56; (d) expand the Yellow District to sector 57, 67, 70, 72, or 59; or (e) expand the Blue District to sector 4, 9, 21, or 12. The Purple District may not be expanded during the next move because its population—352—is already at least 349.
Several of the above moves of type 2 capture one or more sectors including the expansion of the (a) Brown District to sector 30 (which captures sector 16); (b) Red District to sector 40 (which captures sector 41); (c) Red District to sector 38 (which captures sectors 40 and 41); and (d) Orange District to sector 24, 31, 32, 42, 54, or 60. Note that an expansion of the Brown District to sector 9 is not allowed because the Blue District would be trapped (by the Brown District). Also, an expansion of the Yellow District to sector 53, 73, or 74 is not allowed because one or more sectors would be captured and added to the Yellow District, putting its population over the limit of 398.
The next move played is the establishment of the Green District in sector 74. This results in the game position shown in
There are 52 possibilities for the next move which must be of type 1A, 2, or 3. No legal moves of type 3 exist because no district is confined. The feasible moves of type 2 include the 41 moves of type 2 mentioned above and (f) expanding the Green District to sector 70, 75, or 73.
A move of type 1A requires that a new district be established in a sector that is within the most populous connected open space on the board. Among the sectors satisfying this criterion, a sector that ties for being the most steps away from a previously placed home base marker must be selected. In the current board position, the most populous open space has population 511 and consists of sectors 46, 50-51, 56-57, 63-65, 67, 70, and 75. Among these sectors, eight tie for being two steps away from a previously placed home base marker—46, 50-51, 56, 63-65, and 67—and none is three or more steps away from all previously placed home base markers. Thus, there are eight possible moves of type 1A: establish the Gray District in sector 46, 50-51, 56, 63-65, or 67.
The next five moves in this example game are as follows (move type in parentheses):
The establishment of the 8th district during move #5 above ushers in stage 3 of play. During stage 3, the next move must be of type 2 or 3A if at least one type 2 move exists. Otherwise the next move must be of type 2A or 3A. The new board position is shown in
There are 40 possibilities for the next move. Regarding moves of type 2, it is possible to (a) expand the Brown District to sector 5, 13, 22, 21, 19, 50, 46, 28, 29, or 38 (but not 9); (b) expand the Red District to sector 28, 46, 51, 57, 53, 38, or 37; (c) expand the Orange District to sector 18, 65, 63, or 19; (d) expand the Yellow District to sector 57, 67, 70, 72, or 59 (but not 53 or 73); (e) expand the Green District to sector 70, 75, or 73; (f) expand the Blue District to sector 4, 9, 21, 22, 13, or 5; or (g) expand the Gray District to sector 50, 63, 64, 57, or 51. The Purple District may not be expanded during the next move because its population—352—is already at least 349. There are no legal moves of type 3A because no district is confined.
After the next 23 moves in the game, the exemplary board position is shown in
The table below shows the current population of each district. Note that no feasible move of type 2 exists. This is because all districts already have a population of at least 349 and/or are confined. In particular, the only two districts with a population of 348 or less—Blue and Gray—are confined. Thus, the next move must be of type 2A or 3A.
A move of type 3A is possible—i.e. a sector may be reassigned from District X to District Y—if and only if (i) Districts X and Y remain connected after the reassignment, (ii) before the reassignment the population of X exceeds the population of Y by more than the population of the reassigned sector, and (iii) District Y is confined before the reassignment. Note that accurate district population information (as shown in the table above) is needed in order to make a correct assessment regarding item (ii) above.
To search for moves of type 3A, we consider each district one at a time and ask if that district can “steal” a sector from another district.
To search for moves of type 2A, note that the Green District—with a population of 352—is the least populous expandable district. Only one move of type 2A is available: (a) expand the Green District to sector 73. Overall, a total of 14 legal moves exist.
The next five moves are as follows (move type in parentheses):
The board position is now as shown in
Two moves of type 2 are available: (a) expand the Gray District to sector 46 and (b) expand the Brown District to sector 37. Six moves of type 3A are also available: (c) reassign sector 14, 29, or 20 from the Brown District to the Blue District; (d) reassign sector 25 or 31 from the Orange District to the Blue District; and (e) reassign sector 64 from the Green District to the Purple District. Note that the Brown, Red, Yellow, Green, and Gray Districts are not confined, so they may not steal a sector from another district. Also, the Purple District may not steal sector 75 from the Green District because the population imbalance would not be reduced. Overall, eight legal moves are available.
The next five moves are as follows (move type in parentheses):
The new board position is shown in
There are 17 possibilities for the next move. To search for moves of type 2A, note that the Yellow District—with a population of 355—is the least populous expandable district. Only one move of type 2A is available: (a) expand the Yellow District to sector 72. Several moves of type 3A are available: (b) reassign sector 14, 29, or 34 from the Brown District to the Blue District; (c) reassign sector 32 or 33 from the Orange District to the Blue District; (d) reassign sector 64 from the Green District to the Purple District; (e) reassign sector 35 or 50 from the Brown District to the Gray District; (f) reassign sector 36 or 52 from the Red District to the Gray District; (g) reassign sector 55, 62, or 65 from the Orange District to the Gray District; (h) reassign sector 58 from the Yellow District to the Gray District; (i) reassign sector 64 from the Green District to the Gray District; and (j) reassign sector 66 from the Purple District to the Gray District.
The next four moves are as follows (move type in parentheses):
The new board position is shown in
There are 21 possibilities for the next move: (a) reassign sector 5, 13, 21, 22, or 23 from the Blue District to the Brown District; (b) reassign sector 58 or 59 from the Yellow District to the Red District; (c) reassign sector 19 or 31 from the Blue District to the Orange District; (d) reassign sector 68 or 69 from the Yellow District to the Green District; (e) reassign sector 50 from the Brown District to the Gray District; (f) reassign sector 36 or 52 from the Red District to the Gray District; (g) reassign sector 55, 61, or 65 from the Orange District to the Gray District; (h) reassign sector 58 or 68 from the Yellow District to the Gray District; (i) reassign sector 67 from the Green District to the Gray District; or (j) reassign sector 66 from the Purple District to the Gray District.
The next two moves are as follows (move type in parentheses):
The new board position is shown in
There are five possibilities for the next move: (a) reassign sector 31 from the Blue District to the Orange District; (b) reassign sector 50 from the Brown District to the Gray District; (c) reassign sector 36 from the Red District to the Gray District; (d) reassign sector 61 from the Orange District to the Gray District; or (e) reassign sector 67 from the Green District to the Gray District.
The next two moves are as follows (move type in parentheses):
The new board position is shown in
There is only one possibility for the next move: reassign sector 29 from the Brown District to the Blue District. This move is compulsory for the player who takes the next turn.
The new board position is shown in
The final board position at the end of phase 1 is shown in
At the end of phase 1, the scoreboard should read as shown in the box below.
If phase 2 is not played, the game ends, and the Blue Party wins by a score of 5 districts to 3 districts.
If phase 2 is played, an election is simulated. In the final scoreboard (see above), the political margin of the (Brown, Red, Orange, Yellow, Green, Blue, Purple, Gray) District translates to a winning likelihood of (100%, 100%, 100%, 66%, 100%, 100%, 100%, 100%) for the party that has the majority of voters in the district. Note that the Red Party automatically wins three districts with 100% probability and the Blue Party automatically wins four districts.
Dice are then thrown to determine the election results for the district in which there is not an automatic winner. The results are summarized in the table below. Despite being at a disadvantage going into the election, the Red Party “gets lucky” and wins the elections in four districts. The result is a tie.
Strategy games of this Example 29 have a real-world focus in which U.S. congressional districts are formed in a real U.S. state, namely Michigan. These games have taxonomic code A/G/C108/P/3/14/U/2/EXR. These games are analog, multi-player games with complex sectors (which may mimic actual counties of the state) and a political focus in which three types of elements—population, Red Party supporters, and Blue Party supporters—are considered and a pre-determined number of districts are formed. The game proceeds according to alternating, turn-based play; there are 2 players; and moves in categories “E,” “X,” and “R” are allowed.
One example of an initial position is illustrated in
The rules are very similar to Example 28. The main difference is that this game has different population thresholds for moves of type 1 and 2.
All moves made during phase 1 must be of type 1, 1A, 2, 2A, 3, or 3A below.
Phase 1 is divided into three stages. During stage 1, only moves of type 1, 2, and 3 are allowed. Play enters stage 2 if (i) no more moves of type 1 exist and (ii) fewer than 14 districts have been established. During stage 2, only moves of type 1A, 2, and 3 are allowed. Play enters stage 3 immediately after the 14th district is established. During stage 3, the next move must be of type 2 or 3A if at least one move of type 2 exists. Otherwise, the next move must be of type 2A or 3A. This phase of the game ends when no more legal moves exist.
The six types of legal moves are as follows:
We now provide an example of play. After 127 moves have been made—64 by the player representing the Red Party and 63 by the player representing the Blue Party—a final position as shown in
If phase 2 is not played, the game ends, and the Red Party wins by a score of 8 districts to 6 districts.
Strategy games of this example 30 have taxonomic code A/G/C?/N/3/?/U/2-3/EXRBF. They are analog, multi-player, nonpolitical games with turn-based play and complex two-dimensional sectors that are designed for 2-3 players. Each player represents a tribe. The players are provided with a playing surface illustrating a region divided into a number of sectors. The premise is that three tribes have been fighting wars against each other within the region illustrated on the board for more than a century. After significant bloodshed and no clear winner, they have decided to peacefully settle their differences by forming districts that various tribes will inhabit upon conclusion of the game.
During the game, the players organize the region into districts such that (i) each sector is assigned in its entirety to exactly one district and (ii) each district is a single connected piece.
At the outset the districts are formless and the players are informed of each sector's precise shape, location, and set of elements. The set of elements for each sector provides the intensity of each of three elements—rivers, plants, and mammals—in the sector.
During play, the players take alternating turns, with each player taking a single turn before any other player takes another turn. During a player's turn, the player must make one of the following moves: (E) establish a new district by assigning a first sector to it; (X) expand an already established district by assigning a new, previously unassigned sector to the district; (R) reassign a sector from one established district to another established district; (B) break up two adjacent districts by returning all sectors assigned them to unassigned status; and (F) freeze a given district so that no player may modify the district during the next 8 turns. Each player may make each move B and F at most once during the game. Each player has no limit on the number of moves E, X, and R that he/she plays. Each district must be connected at all times during play. In general, any player may use their move to contribute to the construction, destruction, or freezing of any district during any of his/her turns. Play ends when no legal moves exist.
The method of scoring at the end is nontrivial and relates to the suitability (i.e. habitability) of each district for each tribe.
Tribes A, B, and C have different habitability criteria. Members of Tribe A depend on fishing for sustenance and are allergic to plants. That is, Tribe A considers rivers as a resource and plants as a hazard, and it is indifferent to mammals. Members of Tribe B depend on plants/farming for sustenance and are allergic to mammals. That is, Tribe B considers plants as a resource and mammals as a hazard, and it is indifferent to rivers. Members of Tribe C depend on hunting for sustenance and are very poor swimmers. That is, Tribe C considers mammals as a resource and rivers as a hazard, and it is indifferent to plants.
Once the districts are finalized (i.e., when no legal moves exist), the intensity of each element in each district is computed by summing the intensities of the element in the sectors comprising the district.
At the end of the game, each tribe receives points for each district as follows.
Each tribe's point total at the end of the game equals the sum of the points it receives in all districts. The winner is the player (i.e. tribe) with the most points at the end of the game.
Strategy games of this example 31 have taxonomic code A/G/C?/N/6/?/U/2-6/EXRBF. They are analog, multi-player, nonpolitical games with turn-based play and complex two-dimensional sectors that are designed for 2-6 players. Each player represents an interplanetary transportation company. In one example, the premise is that six interplanetary transportation companies dominate the economy of the region of the Milky Way Galaxy in the year 2388. After decades of chaos in the transportation market, the companies have decided, for their mutual benefit, to set standard transportation rates within the galaxy by dividing it into districts. After the districts are formed, direct transportation between planets will only take place (i) within districts and (ii) between adjacent districts. No other direct transportation services will be offered.
During the game, players organize the galaxy—which is already divided into C sectors that (i) may not be further divided, (ii) do not overlap, and (iii) together cover the entire galaxy—into a given number, D, of districts (where 2≤D<C) such that (a) each sector belongs in its entirety to exactly one district and (b) each district is comprised of a set of adjacent sectors.
At the outset the districts are formless and the players are informed of each sector's precise shape, location, and set of elements, which in this case is a set of planets. The set of planets in a sector consists of six numbers which respectively represent the number of each planet type—agricultural, metropolitan, scholarly, industrial, medical, and ecological—in the sector.
Players take turns in rotating fashion. During a player's turn, the player must make one of the following moves: (E) establish a new district by assigning a first sector to it; (X) expand an already established district by assigning a new, previously unassigned sector to the district; (R) reassign a sector from one established district to another established district; (B) break up two or three adjacent districts by returning all sectors previously assigned them to unassigned status; and (F) freeze a given district so that no player may modify the district during the next 5 turns. Each player may make each move B and F at most once during the game. Each player has no limit on the number of moves E, X, and R that he/she plays. Each district must be connected at all times during play. In general, any player may use their move to contribute to the construction, destruction, or freezing of any district during any of his/her turns. Play ends when no legal moves exist.
The method of scoring is nontrivial; it relates to the profitability of each district and each pair of adjacent districts for each company.
Each company specializes in a different kind of transportation and therefore has a different perspective on profitability. Company A specializes in transporting food and food equipment between agricultural and metropolitan planets. Company B specializes in transporting students and researchers between metropolitan and scholarly planets. Company C specializes in transporting workers and engineers between scholarly and industrial planets. Company D specializes in transporting injured workers between industrial and medical planets. Company E specializes in transporting people and medicinal plants between medical and ecological planets. Company F specializes in transporting flora and fauna between ecological and agricultural planets.
Once the districts are finalized (i.e., once no legal moves exist), the total number of each planet type in each district is computed by summing the number of that planet type in the sectors comprising the district. Each company receives points for each district as follows.
Each company also receives points for each pair of adjacent districts (e.g. X and Y) as follows.
Each company's point total at the end of the game equals the sum of the points it receives in all districts plus the sum of the points it receives in all pairs of adjacent districts. The winner is the player (i.e. company) with the most points at the end of the game.
From the foregoing, it will be appreciated that although specific examples have been described herein for purposes of illustration, various modifications may be made without deviating from the spirit or scope of this disclosure. For example, even though the Examples are described as being analog, they could alternatively be digital, and any component of the games could be digitally represented. It is therefore intended that the foregoing detailed description be regarded as illustrative rather than limiting, and that it be understood that it is the following claims, including all equivalents, that are intended to particularly point out and distinctly claim the claimed subject matter.
This application claims priority to U.S. Provisional Application Ser. No. 62/785,824, filed on Dec. 28, 2018, the entirety of which is incorporated by reference herein.
| Number | Date | Country | |
|---|---|---|---|
| 62785824 | Dec 2018 | US |
