Patent Application
20180361228
The present invention, in some embodiments thereof, relates to a three-dimensional game or puzzle and, more particularly, but not exclusively, to a game or puzzle based on a cube or a pyramid.
A MAGIC SQUARE is an n×n square subdivided into n2 cells inscribed with disjoint (distinct) integers arranged in a way such that the sums of the integers in all the rows and all the columns and the two diagonals are equal to the same constant which is called the MAGIC CONSTANT. An example known in the literature as the traditional LO SHU Chinese Magic square, is shown in
The magic Constant of the Lo Shu Magic Square is 15 which happens to be the number of days in each of the 24 cycles (or terms) of the Chinese Solar year. The Lo Shu magic square is mentioned in a Chinese legend dated several centuries before the Christian era. Since then and during the centuries magic squares have been studied by hobbyists, Astrologists, alchemists and by mathematicians. Magic squares have been used as Talismans and Amulets and stirred the imagination of recreational mathematicians. They have been generalized in many ways: multi-dimensional, with words or geometric figures instead of numbers, with many rows and columns etc.
There are more than 300 papers in the literature on recreational mathematics on the subject of magic squares.
U.S. Pat. No. 7,644,924 shows a cube that is designed for a game where numbers are placed on each face of the cube to form a soduko square. These squares are completely independent, with no linking properties between faces of the cube or between numbers on each face.
U.S. Pat. No. 7,100,917 and U.S. Pat. No. 6,626,431 show 3 dimensional Rubiks cube based games with magic squares on each face of the cube. These squares are completely independent with no linking properties
U.S. Pat. No. 9,492,734 discloses multi-dimensional puzzles for teaching, entertainment, and construction. One embodiment comprises two forms, one planar and one non-planar, which comprise a plurality of protrusions and recesses. These forms may be assembled by selectively interconnecting the various protrusions and recesses to create multi-dimensional shapes such as a cube, cylinder, and sphere among many other possibilities.
The possibility to transform the star of David into a magic structure with a magic constant is known in the literature. The shape is static, and the integers are fixed along the star. One specific magic structure star is known in the art, in which the integers used are the consecutive integers 1 to 12 and the magic constant is 26.
The present embodiments may provide cubes with 6 different 3×3 squares, where these squares have a common magic constant n in a way such that a linking magic property between neighboring squares is provided, and also pyramids and stars of David. Individual numbers are removed to provide a challenge of recreating the shape with the original conditions.
According to an aspect of some embodiments of the present invention there is provided a three-dimensional object for use as a puzzle, the object having a plurality of faces, each face having a plurality of locations and numbers at each location respectively, wherein at least some of the numbers are replaceably removable, remaining numbers being fixed, and a predetermined rule links all of said numbers on the plurality of faces to enable correct replacement of said removed numbers to solve said puzzle.
In an embodiment, said locations comprise a center location, edge locations and corner locations and numbers at each location.
In an embodiment, said removable numbers are mounted on buttons.
In an embodiment, said buttons are magnetic.
In an embodiment, said buttons are clip-on buttons.
In an embodiment, said buttons are mounted on rotatable levers.
The object may be a cube.
In an embodiment, each face of the cube has nine numbers arranged in three rows of three.
In an embodiment, each face is a magic square having a pivot number, and said rule defines a common pivot number for respective faces.
In an embodiment, each magic square is a non-negative magic square.
An embodiment may have eight corner triplets, each one of said eight corner triplets summing to three times said common pivot number.
An embodiment may have six middle hoop 4 tuples, each of said six middle hoop 4 tuples summing to a common constant.
An embodiment may have eighteen diagonally opposite sub corner 4 tuples, said diagonally opposite sub corner 4 tuples summing to one constant.
An embodiment may have six opposite middle 4 tuples, said six opposite middle 4 tuples summing to a common constant.
In an embodiment, said common constant is four times said common pivot.
An embodiment may have eight diagonal ring six tuples, each of said eight diagonal ring six tuples summing to a second common constant.
In an embodiment, said second common constant is six times said common pivot.
In an embodiment, said rule defines first and second common pivots applied to different ones of said faces, each face comprising a non-negative magic square with a respective one of said common pivots, the cube having eight corner triplets, each one of said corner triplets summing to a respective constant which is related to one of said common pivots.
In an embodiment, said respective constant is three times said related one of said common pivots.
A cube having faces, may comprise nine numbers per face arranged as three rows in three columns, all faces having a common pivot. One row and one column may be removed from each face to form respective reduced faces having two rows of two columns, each reduced face being nonnegative, having four integers which are distinct. The reduced faces are not equivalent to one another, and respective diagonals of said reduced faces sum to twice the common pivot, the reduced faces having eight corner triplets summing to a first constant, the reduced faces having eighteen diagonally opposite sub corner 4 tuples, the diagonally opposite sub corner 4 tuples, and the faces together summing to a second constant.
In an embodiment, the first constant is three times the common pivot and/or the second constant is four times the common pivot.
The object may alternatively be a pyramid having three or four triangular faces.
In an embodiment, each of said triangular faces has six positive numbers arranged in triangular form as three edges of three numbers with shared vertices, each triangle being non-equivalent, said rule defining that all numbers in a first three of said faces are respectively distinct.
In an embodiment, said rule defines that one of said faces includes three distinct pairs of equal numbers.
In an embodiment, all the edges of said pyramid sum to a first common constant.
In an embodiment, said first common constant is three times a pivot number, said pivot number being present in each face.
In an embodiment, the numbers of the three middle rows of said first three faces extend as a belt around said pyramid sum to a second constant.
In an embodiment, said second constant is six times a pivot number, said pivot number being present in each face.
According to a second aspect of the present invention there is provided a pair of the pyramids wherein the pyramids share a common pivot number. The pyramids may be combined by attaching two faces, such that numbers align along edges between respective pyramids of said pair to form eight four-tuples, each four tuple comprising two numbers from a first pyramid of said pair and two numbers from a second pyramid of said pair, wherein the numbers of each four tuple sum to a common constant.
In an embodiment, said common constant is four times said common pivot number.
According to a third aspect of the present invention there is provided a three-dimensional object for use as an ornament, the object having a plurality of faces, each face having a plurality of locations and numbers at each location respectively, wherein said locations comprise a center location, edge locations and corner locations and numbers at each location, said numbers being selected such that a predetermined rule links all of said numbers on the plurality of faces.
According to a fourth aspect of the present invention there is provided a three-dimensional object forming a magic cube, the object having a plurality of faces, each face having a plurality of locations and numbers at each location respectively, wherein said locations comprise a center location, edge locations and corner locations and numbers at each location, said numbers being selected such that a predetermined rule links all of said numbers on the plurality of faces.
According to a fifth aspect of the present invention there is provided a three-dimensional object forming a pyramid, the object having a plurality of faces, each face having a plurality of locations and numbers at each location respectively, and a predetermined rule links all of said numbers on the plurality of faces to enable correct replacement of said removed numbers.
According to a sixth aspect of the present invention there is provided a three-dimensional object forming a cube, the object having a plurality of faces, each face having a plurality of locations and numbers at each location respectively, wherein said locations comprise a center location, edge locations and corner locations and numbers at each location, said numbers being such that a predetermined rule links all of said numbers on the plurality of faces, all faces having a common pivot number, wherein one row and one column is removed from each face to form respective reduced faces having two rows of two columns, each reduced face being nonnegative, having four integers which are distinct, wherein respective reduced faces are not equivalent to one another, and wherein respective diagonals of said reduced faces sum to twice the common pivot.
According to a seventh aspect of the present invention there is provided a star of David comprising two interlinked oppositely facing triangles, each triangle having three vertices and edges of said triangles meeting at six intersections, the star of David having a number located at each vertex and each intersection, to form six straight lines of four numbers respectively, and each number being common to two of said lines, the numbers of each line summing to a common constant, wherein at least some of the numbers are replaceably removable to form a puzzle.
In an embodiment, some of said numbers are greater than 12.
The objects may be provided as or incorporated into an amulet, an item of jewelry, a wall decoration, an item of desk furniture and an educational toy.
According to an eighth aspect of the present invention there is provided a star of David comprising two interlinked oppositely facing triangles, each triangle having three vertices and edges of said triangles meeting at six intersections, the star of David having a number located at each vertex and each intersection, to form six straight lines of four numbers respectively, and each number being common to two of said lines, the numbers of each line summing to a common constant, and wherein a plurality of said numbers is greater than 12.
According to a ninth aspect of the present invention there is provided a three-dimensional object comprising a plurality of faces, each face having corners and sharing edges and having adjacent corners with another face and the object having vertices, each vertex having a plurality of vertex numbers located thereon, one of said vertex numbers being on each of the faces of said vertex and each corner having a number located thereon, the numbers being placed according to a rule wherein all of the numbers placed on the vertices of the object sum to a first predetermined number, and all the numbers placed on the corners of the object amount to a second predetermined number, and a sum of said first predetermined number and said second predetermined number is a multiple of a third predetermined number.
In an embodiment, additional predetermined rules link all of said numbers on the plurality of faces.
The object may be a tetrahedron, a cube a double pyramid, a double tetrahedron, an octahedron, a dodecahedron, and an icosahedron, or may be a planar representation thereof.
As above, one or more of the numbers may be removed from a respective vertex or a face of the object, either to make a new, derivative object that retains the original rule, or to create a puzzle by using the rules to replace the missing numbers.
Unless otherwise defined, all technical and/or scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which the invention pertains. Although methods and materials similar or equivalent to those described herein can be used in the practice or testing of embodiments of the invention, exemplary methods and/or materials are described below. In case of conflict, the patent specification, including definitions, will control. In addition, the materials, methods, and examples are illustrative only and are not intended to be necessarily limiting.
Implementation of the method and/or system of embodiments of the invention can involve providing physical shapes into which numbers are inserted, or a virtual shape on a computer screen on which the same tasks are carried out, thereby performing or completing selected tasks manually, automatically, or a combination thereof. Moreover, according to actual instrumentation and equipment of embodiments of the method and/or system of the invention, several selected tasks could be implemented by hardware, by software or by firmware or by a combination thereof using an operating system.
For example, hardware for performing selected tasks according to embodiments of the invention could be implemented as a chip or a circuit. As software, selected tasks according to embodiments of the invention could be implemented as a plurality of software instructions being executed by a computer using any suitable operating system. In an exemplary embodiment of the invention, one or more tasks according to exemplary embodiments of method and/or system as described herein are performed by a data processor, such as a computing platform for executing a plurality of instructions.
Optionally, the data processor includes a volatile memory for storing instructions and/or data and/or a non-volatile storage, for example, a magnetic hard-disk and/or removable media, for storing instructions and/or data. Optionally, a network connection is provided as well. A display and/or a user input device such as a keyboard or mouse are optionally provided as well.
Some embodiments of the invention are herein described, by way of example only, with reference to the accompanying drawings. With specific reference now to the drawings in detail, it is stressed that the particulars shown are by way of example and for purposes of illustrative discussion of embodiments of the invention. In this regard, the description taken with the drawings makes apparent to those skilled in the art how embodiments of the invention may be practiced.
In the drawings:
The present invention, in some embodiments thereof, relates to a three-dimensional ornament, game or puzzle and, more particularly, but not exclusively, to an ornament game or puzzle based on a cube or a pyramid or a star shape or a tetrahedron etc.
The puzzles all have rules for completion, and the rules for any given embodiment may make use of rules including those in the following definitions.
Before explaining at least one embodiment of the invention in detail, it is to be understood that the invention is not necessarily limited in its application to the details of construction and the arrangement of the components and/or methods set forth in the following description and/or illustrated in the drawings and/or the Examples. The invention is capable of other embodiments or of being practiced or carried out in various ways.
Referring now to the drawings,
The locations at which numbers are placed may include center locations, edge locations and corner locations. In a square face, the numbers are typically placed in rows and columns. In a pyramid the numbers are typically placed along the edges.
The removable numbers may be mounted on buttons. The buttons may be magnetic or clip on or use other mechanical means to attach and detach. In another embodiment, where the numbers rotate, the buttons may be mounted on rotatable levers, as with the arrangement of the Rubic cube.
b, as discussed in greater detail hereinbelow, illustrate various embodiments in which the object is a cube. The cube may have nine numbers arranged in three rows of three on each one of six faces. Each face may define a magic square having a pivot number, and the rule may define a common pivot number for two or more of the different faces. In one embodiment, the same pivot number applies to all six faces. In embodiments each magic square in the cube is a non-negative magic square.
The cube may have eight corner triplets, and each of the eight corner triplets may sum to a common number, which may in an embodiment be three times the common pivot number.
The cube may have six middle hoop 4 tuples, which sum to a common constant.
The cube may have eighteen diagonally opposite sub corner 4 tuples, which may sum to one constant.
The cube may have six opposite middle 4 tuples which sum to a common constant.
In the above case, the common constant may be four times the common pivot.
The cube may have eight diagonal ring six tuples, each of which sum to a second common constant. The second common constant may be six times the common pivot.
An embodiment is discussed in greater detail below in respect of
In one of the embodiments, as discussed in greater detail below in respect of
In an embodiment of such a two-by-two face cube, the first constant may be three times the common pivot and/or the second constant may be four times the common pivot.
As shown in
The edges of the pyramid across all the faces may each sum to a first common constant.
That common constant may be three times a pivot number, and the pivot number may be present in each face.
An additional rule concerns a belt around the middle of the upper faces of the pyramid. The numbers of the three middle rows of the upper faces may be considered to form a belt around the pyramid, and the numbers in the belt may sum to a specific constant.
That specific constant may for example be six times the pivot number.
As shown in
The two pyramids of the pair may be placed or attached together via respective faces, so that the numbers in facing edges of the two pyramids can be paired together. When so doing, the numbers align along the edges to form eight four-tuples where each four tuple comprises two numbers from each pyramid. The numbers of each four tuple sum to a common constant, say four times the common pivot number.
Reference is now made to
As shown in the figures, two or more of the numbers may be greater than 12 in order to increase the challenge.
The three-dimensional shape whether a cube, or a pyramid or a pair of pyramids or the star of David may be an amulet, an item of jewelry, a wall decoration, an item of desk furniture or toy, such as an educational toy.
The linked magic cube.
The Lucas Formula, and Magic Square Properties
The concept of magic squares and in particular, 3×3 magic squares, is now considered in greater detail.
For such 3×3 squares a mathematician, Eduard Lucas, developed in the 19th century, a general formula, shown in
Where:
For positive squares the following inequalities must hold:
0<a<b<(c−a)
b is not equal to 2a.
For non negative squares the following inequalities must hold:
0<a<b</=(c−a)
b is not equal to 2a.
In addition to the fact that the integers in the square are distinct. It follows from the formula in
It can be shown that the number N of non equivalent magic squares where:
the pivot is equal to a given c:
a is the biggest integer,
a<(c−1)/3
N=((c−1)/2)(c−3)/2))−a
N=(c/2−1)2−a
N=(c−1)2/4−a
N=((c/2)−1))(c/2)−a
It follows from the above formulas that the number of cubes (x), with six faces comprising non equivalent magic squares having the same pivot c, is calculated as:
X=((M(c)(M(c)−1) . . . (M(c)−5))*(86).
where M(c) is the number of non equivalent magic squares with their pivot equal to c.
It is easy to show that for c as small as 10, the number of cubes gets bigger than 2 trillion (1012), a very large number. This implies that if we want to find a cube with special properties among all the cubes defined above, an exhaustive search is not practical.
There are only specific cubes with magic squares which hold the properties of WONDER CUBES, (linked magic squares on a cube) where the pivot=c. These cubes have the following properties:
Quasi wonder cubes are also specific cubes with magic squares, with the following properties, where pivots=c,d and all the magic squares on its faces are non negative and are not equivalent to one another
Following removal of all the 24 corner cells from a cube, the result will be a cube whose six faces are cross shaped. As an alternative, one may remove from the wonder cube the middle rows and the middle columns of each of the six faces, resulting in a cube whose faces are 2×2 squares, thus providing a 2×2×3 cube. The wonder properties of the wonder cube are distributed between those two solid bodies while all the integers on the wonder cube are distributed between those two bodies.
The Cube whose Six Faces are Cross Shaped.
It is easy to verify that this cube inherits the following properties from the wonder cube source:
It is then possible to reconstruct the source wonder cube from the derived cube by using the half diagonal property, which is also inherited by the derived cube. By this property every missing corner in the derived cube is equal to half the sum of the two integers forming the half diagonal on the other side of the main diagonal facing the corner.
2×2×3 Cubes Derived from 3×3×3 Wonder Cubes.
If we remove the middle rows and columns from all the six squares on a 3×3×3 cube we get a 2×2×3 cube whose faces are made of 2×2 squares. The derived 2×2 squares, are nonnegative, their 4 integers are distinct and are not equivalent to one another. It is well known and easy to prove that 2×2 squares cannot have the normal properties of magic squares, however, the diagonals of those 2×2 squares sum to 2c, where c is the pivot of the original 3×3 squares and some of the linking properties of the 3×3×3 cube are inherited by the 2×2×3 cube:
In the following we show how to transform a wonder cube into two different bodies.
Note that:
The second layer includes the six Integers around the three integers in the first layer and they sum to 6×11=66. Similarly the third layer includes 9 integers that sum to 9×11=99. The fourth layer includes again six integers that sum to 6×11=66 and the fifth and last layer includes 3 integers, 10,9 and 14 that sum to 3×11=33.
The three parallelograms in
Dual platonic solids are described in Wikipedia as follows:
Two platonic solids are dual if they have the same number of edges and the number of faces of one is equal to the number of vertices of the other.
The tetrahedron as discussed above is self dual since the number of edges (4) is equal to the number of faces (4). The cube is dual to the octahedron as the number of edges of both solids is 12 while the number of vertices of the cube is 8, which is equal to the number of faces of the octahedron, and the number of faces of the cube is 6, which is equal to the number of vertices of the tetrahedron. In the same way, the two solids, the dodecahedron and the icosahedron are dual solids. Herein we extend the geometric duality of the cube and the octahedron to a same sum duality. The cube in this extension is a 2×2×3 cube. This extension is illustrated in
In greater detail,
A double tetrahedron is a pair of tetrahedrons set one over the other wherein one of the two tetrahedrons is inverted so that the top triangle of one is connected to the bottom triangle of the other tetrahedron.
The procedure of creating two TWIN double tetrahedrons is illustrated in
Star of David Shapes with Number Sets on each Triangle that have Linked Properties
The numbers on the magic squares with linked properties can be re arranged to be set in the shape of numbers placed on triangles making the shape of the Star of David. Numbers on each Star of David are placed around the pivot and have magic square properties, these are linked to one or more Stars of David placed in the same planar dimension.
More particularly, if we join the planar layout of the two parts of the cube, shown in
We now consider the shield of David same sum aspects. It is possible to attach the numbers 1 to 12 to the six vertices of the two triangles and the six intersections of the triangles in a way such that the sum of the four numbers along any of the six edges of the shield sum to the integer 26. It is easy to prove and left to the reader, that it is not possible to construct a Star of David endowed with twelve distinct integers, whose magic constant is less than 26. It is however possible to replace the 12 integers 1 to 12 by twelve other different integers in such a way that the four numbers along any edge of the two triangles sum to any constant greater or equal to 27. We may call this constant the MAGIC CONSTANT as before. In a variant, the integer 10 is replaced by 13 and has magic constant equal to 27. As for magic constant greater or equal to 28 we find a formula which enables the construction of shields of David with magic constant greater or equal to 28. We note:
This is an example of a wonder cube with numbers significant to the Chinese tradition. Its pivot is equal to 8 and its magic constant is equal to 3*8=24, as the number of cycles in the Chinese solar year and it includes in its squares all numbers from 1 to 15, as the number of days in every cycle. One can also find a row in one of the squares that contain the numbers 8 and 4 which hints at the birth date of Buda, celebrated in China in the eight day of the fourth Chinese month.
This wonder cube has Christian religious significance: 33 is the age of Jesus at his crucifixion according to the Christian tradition. The number 3, as well as the three dimensionality of the object hint to the Christian holy trinity. The cube has 12 edges as the number of Jesus disciples or apostles that were present at the last supper. 44 is a double of the number 4 which is the number of the creation, particularly the earth. In addition, the final part of Jesus' earthly ministry is 44 days long. This period begin on the day he was crucified, and ended on the day he gave his final instructions to his disciples and ascended to heaven from the Mount of Olives. Lastly, The Vatican is thought to occupy 44 hectares in the city of Rome. Lastly, most Protestant Christians recognize 66 books in the Bible (66 books in the King James Bible)
This exemplary wonder cube has Jewish implications. The pivot of all its six squares is equal to 12, the number of the tribes of Israel. The number 36, in the Jewish tradition, is the number of just men that exist in the world and justify the existence of the world. The dates of many Jewish holidays (when Jewish months are counted by numbers) can also be found in some squares on the cube.
The cube has the inherited property of the sum of the eight corner triplets which is 30; and the 18 diagonally opposite sub corner 4 tuples, and the faces of each cube, sum 40.
The cube has the inherited property of the sum of the eight corner triplets which is 24, and the 18 diagonally opposite sub corner 4 tuples, and the faces of each cube, sum to 32.
The cube has the inherited property of the sum of the eight corner triplets which is 33, and the 18 diagonally opposite sub corner 4 tuples, and the faces of each cube, sum to 44.
The cube has the inherited property of the sum of the eight corner triplets which is 36, a number with Jewish traditional implications and the 18 diagonally opposite sub corner 4 tuples, and the faces of each cube, sum to 48.
The cube has the inherited property of the sum of the eight corner triplets which is 48 and the 18 diagonally opposite sub corner 4 tuples, and the faces of each cube, sum to 64
The cube has the inherited property of the sum of the eight corner triplets which is 54 and the 18 diagonally opposite sub corner 4 tuples, and the faces of each cube, sum to 72.
The cube has the inherited property of the eight corner triplets, which split into 2 sets, one set which sums to 15 and the other set sums to 24; and the 18 diagonally opposite sub corner 4 tuples, and the faces of each cube, sum to 20 and 32
The cube has the inherited property of the eight corner triplets, all which sum to 32; and the 18 diagonally opposite sub corner 4 tuples, and the faces of each cube, sum to 32 and 48.
Specific Examples of Puzzle Games on Wonder Cubes and Variations.
All games can be implemented as three dimensional tangible games using physical cubes or as three-dimensional rotatable figures which are part of a computer or smartphone game. Parts of the cube can be attached and detached from the six faces of the cube by magnetic means or mechanical means (e.g. a button is set on each side of the plate and a receptor button on each face of the cube), or via a computer based animation.
GAME: Remove from the cube all the 24 corner integers or a subset of them. Scramble them then reset them on the cube so as to restore the original cube back to its linked properties. The more cells removed the more difficult the reconstruction. The player choses how hard the challenge.
The derived cube inherits some of the properties from the original cube, e.g. its corner triplets sum to 3c (24). The numbers on each face sum to 32. The derived cube is constructed in a way such that its faces can be removed and reset on the cube. When removed both sides of the faces include the same 4 integers but the arrangement of the integers on one side is reflected image of the arrangement on the other side. The player may remove one to five faces, turn over some of the removed faces, scramble the faces and then reset them on the cube in such a way as to get back the original cube. The more removed faces the greater the challenge.
This is a puzzle that can be printed in a newspaper (as the Sudoku puzzle). The figure represents a spread out of a 2×2×3 cube derived from the 3×3×3 cube with pivot equal to 8. To solve the puzzle one has to insert positive integers into the empty cells under the following conditions: The four integers in any of the six squares are distinct and the two diagonals in the square sum to the constant 16, no square can be derived from another square by rotation or reflection and the eight corner triplets of the cube represented in the figure sum to the constant 24. The three cells forming a corner triplet are marked in the figure by the same letter A to H, to help the person trying to solve the puzzle. One can design many different such puzzles and their solution is not unique.
This puzzle is far from trivial since the number of ways to enter the integers into the empty cells under the given conditions apart from the condition on the corner triplets is very large so that an extensive search is not practical and one has to use heuristic methods to solve it. However the puzzle can be set in many ways so as to reduce complexity by increasing the number of integers which are given, and inserting them in additional locations.
Reference is now made to
One can pair an edge from the left pyramid 240 to an edge from the right pyramid 242 so that 4 integers, two on both sides of the left edge plus the corresponding two on both sides of the edge on the right sum to 44.
The player should find the matching of the edges.
The pyramid is another derivation of the linked magic square.
From every square, 3 pairs of pyramids may be derived, each pair with linking properties.
The pyramid has the usual magic properties i.e. every one of its 4 faces includes
The two pyramids may be arranged such that one face is placed on a face of the other pyramid.
There are 64 ways to combine the two pyramids together, and of these, some combinations may cause the two pyramids to have linked properties, where corresponding pairs of integers sum to a magic number.
In the example: The 2 edges (U1,Q0 and (Q,U3) in
One Combination will cause the following 4 tuples: (10,7)(20,7); (15.10)(1,18); (8,16)(10,8). Of the 64 optional combinations, some will cause the pyramids to have linked properties between 4 tuples of corresponding pairs.
In a specific case, one may place the two pyramids such that one of the bases is placed on the base of the other pyramid. The result is to form one object comprised of the two pyramids such that the bases are hidden. There is one way, from three options, in which the combined integers on the edge of the double pyramid object may cause the two pyramids to have linked properties.
The following linked properties exist between the twin pyramids placed correctly:
Reference is now made to
Pyramids, magic Stars of David and magic cubes can be designed as ornaments to be placed on a desk, or hung on a wall. Likewise, pyramids, magic Stars of David and magic cubes can be designed as amulets or as charms to be added to a neckless or bracelet, such as a Pandora bracelet or necklace. U.S. Pat. No. 7,007,507 discloses Pandora bracelets.
For example, The twin pyramids can be used as amulets that can be given to two separate people, symbolizing the connection between them with a specific number which is a part of the linking properties of the pyramids. (for example: c; 4c or 6c)
The magic constant in
Reference is now made to
Reference is now made to
Reference is now made to
Reference is now made to
The various shapes discussed herein, including platonic solids and stars, may be used as a game that may also fit a computer or smartphone game. In the game, some of the integers are removed from the stars and the player is required to find the removed integers. The more integers removed the harder the challenge.
It is expected that during the life of a patent maturing from this application many relevant three-dimensional shape technologies and shape representation technologies will be developed and the scopes of the corresponding terms are intended to include all such new technologies a priori.
The terms “comprises”, “comprising”, “includes”, “including”, “having” and their conjugates mean “including but not limited to”.
The term “consisting of” means “including and limited to”.
The term “consisting essentially of” means that the composition, method or structure.
As used herein, the singular form “a”, “an” and “the” include plural references unless the context clearly dictates otherwise.
It is appreciated that certain features of the invention, which are, for clarity, described in the context of separate embodiments, may also be provided in combination in a single embodiment. Conversely, various features of the invention, which are, for brevity, described in the context of a single embodiment, may also be provided separately or in any suitable subcombination or as suitable in any other described embodiment of the invention and the description herein is to be construed as if these embodiments, combinations and sub-combinations are explicitly set forth herein. Certain features described in the context of various embodiments are not to be considered essential features of those embodiments, unless the embodiment is inoperative without those elements.
Although the invention has been described in conjunction with specific embodiments thereof, it is evident that many alternatives, modifications and variations will be apparent to those skilled in the art. Accordingly, it is intended to embrace all such alternatives, modifications and variations that fall within the spirit and broad scope of the appended claims.
All publications, patents and patent applications mentioned in this specification are herein incorporated in their entirety by reference into the specification, to the same extent as if each individual publication, patent or patent application was specifically and individually indicated to be incorporated herein by reference. In addition, citation or identification of any reference in this application shall not be construed as an admission that such reference is available as prior art to the present invention. To the extent that section headings are used, they should not be construed as necessarily limiting.
This application claims the benefit of priority under 35 USC § 119(e) of U.S. Provisional Patent Application No. 62/520,076, filed on Jun. 15, 2017. The contents of the above applications are all incorporated by reference as if fully set forth herein in their entirety.
| Number | Date | Country | |
|---|---|---|---|
| 62520076 | Jun 2017 | US |
