US20180361228A1

US20180361228A1 - Three dimensional games and puzzles

Info

Publication number: US20180361228A1
Application number: US16/009,325
Authority: US
Inventors: Azaria PAZ
Original assignee: Individual
Current assignee: Individual
Priority date: 2017-06-15
Filing date: 2018-06-15
Publication date: 2018-12-20

Abstract

A three-dimensional object for use as a puzzle, has a plurality of faces, each face having a locations and numbers at each location. Some of the numbers are replaceably removable, remaining numbers being fixed, and a predetermined rule links all of said numbers on the different faces to enable correct replacement of the removed numbers to solve said puzzle.

Description

RELATED APPLICATIONS

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.

FIELD AND BACKGROUND OF THE INVENTION

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 n²cells 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 FIG. 1.
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.

SUMMARY OF THE INVENTION

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.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

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:

FIG. 1 shows an ancient magic square known from China;

FIGS. 2a to 2i illustrate cubes according to embodiments of the present invention;

FIGS. 3a to 3f illustrate opened out cubes of the present embodiments and illustrate various definitions of rule parts for solving the puzzle according to embodiments of the present invention;

FIG. 4 illustrates a general formula for a magic square;

FIG. 5 is a schematic illustration of an exemplary non negative wonder cube where the pivot is equal to 10 and the magic constant of the squares is 30, according to embodiments of the present invention;

FIG. 6 is a schematic illustration of an exemplary positive wonder cube, with the pivot equal to 8 according to embodiments of the present invention;

FIG. 7 is a schematic illustration of an exemplary positive wonder cube with the pivot equal to 11 according to embodiments of the present invention;

FIG. 8 is a schematic illustration of an exemplary positive wonder cube with the pivot equal to 12 according to embodiments of the present invention;

FIG. 9 is a schematic illustration of an exemplary positive wonder cube with the pivot equal to 16 according to embodiments of the present invention;

FIG. 10 is a schematic illustration of an exemplary positive wonder cube with the pivot equal to 18 according to embodiments of the present invention;

FIG. 11 is a schematic illustration of an exemplary quasi wonder cube with the pivots equal to 5 and 8 according to embodiments of the present invention;

FIG. 12a is a schematic illustration of an exemplary quasi wonder cube with the pivots equal to 8 and 12 according to embodiments of the present invention;

FIG. 12b is a schematic illustration showing how an exemplary 2×2×3 cube is derived from a 3×3×3 wondercube according to embodiments of the present invention;

FIGS. 13a-c are schematic illustrations of an exemplary 2×2×3 cube derived from a wonder cube with the pivot equal to 10 according to embodiments of the present invention;

FIG. 14 is a schematic illustration of an exemplary 2×2×3 cube derived from a wonder cube with the pivot equal to 8 according to embodiments of the present invention;

FIG. 15 is a schematic illustration of an exemplary 2×2×3 cube derived from a wonder cube with the pivot equal to 11, according to embodiments of the present invention;

FIG. 16 is a schematic illustration of an exemplary 2×2×3 cube derived from a wonder cube with the pivots equal to 12, according to embodiments of the present invention;

FIG. 17 is a schematic illustration of an exemplary 2×2×3 cube derived from a wonder cube with the pivot equal to 16, according to embodiments of the present invention;

FIG. 18 is a schematic illustration of an exemplary 2×2×3 cube derived from a wonder cube with the pivots equal to 18, according to embodiments of the present invention;

FIG. 19 is a schematic illustration of an exemplary 2×2×3 cube derived from a quasi wonder cube with the pivots equal to 5 and 8, according to embodiments of the present invention;

FIG. 20 is a schematic illustration of an exemplary 2×2×3 cube derived from a quasi wonder cube with the pivots equal to 8 and 12, according to embodiments of the present invention;

FIG. 21a is a photograph of an example of a puzzle game based on the exemplary cube with c=11, according to embodiments of the present invention;

FIGS. 21b and c are examples of a puzzle game based on the exemplary cube with p=11, according to embodiments of the present invention;

FIG. 22a is a photograph of an exemplary derived 2×2×3 reduced cube derived from a wonder cube with P=12, according to embodiments of the present invention;

FIG. 22b is a photograph of an exemplary derived 2×2×3 reduced cube made from a wonder cube with p=8, according to embodiments of the present invention;

FIGS. 23a and 23b are photographs of the planar spread out of an exemplary cube derived from a wonder cube with p=8, according to embodiments of the present invention;

FIG. 24 is a photograph showing an exemplary pair of magic pyramids according to embodiments of the present invention;

FIG. 25 is a schematic illustration of a planar layout of a magic pyramid derived from a square with a pivot of C=11, according to embodiments of the present invention;

FIG. 26 is a schematic illustration of a planar layout of the twin pyramid 242 also derived from the square with pivot c=11, according to embodiments of the present invention;

FIGS. 27, 28. 29, 30 and 31 are schematic illustrations and a photograph showing various magic stars of David, according to embodiments of the present invention;

FIG. 32 is a photograph illustrating a layout for a 2×2×3 wonder cube with a hidden pivot equal to 10 according to an embodiment of the present invention;

FIG. 33 is a photograph showing a representation of a layout for an octahedron according to embodiments of the present invention;

FIGS. 34 and 35 are photographs respectively showing a cube and a planar layout giving a part of a star according to embodiments of the present invention;

FIG. 36 illustrates a star formed by joining the parts of FIGS. 34 and 35;

FIG. 37 is a simplified diagram illustrating twin tetrahedrons according to an embodiment of the present invention;

FIG. 38 is a schematic diagram which illustrates four platonic solids, and shows one way in which numbers may be distrusted on the solids to provide embodiments of the present invention;

FIG. 39 illustrates upper and lower parts of a dodecahedron in planar form, which may be cut out and folded to fit together;

FIG. 40 illustrates a cube with pivot 11 according to embodiments of the present invention; and

FIG. 41 illustrates upper and lower parts of an icosahedron in planar form for cutting out and folding.

DESCRIPTION OF SPECIFIC EMBODIMENTS OF THE INVENTION

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.

Definitions:

- 1. A WONDER OBJECT is a 3 dimensional object in which all of the numbers placed on the vertices of the object sum to one same number and all the numbers placed on the corners of the faces of the object amount to one same number, and the vertex sum and the sum of the numbers on each of the corners of the faces are multiples of one constant number. That constant number is referred to herein as the pivot constant—see the following definition.
- 2. The PIVOT CONSTANT is an anchor number of which multiplications derive the predefined rules linking numbers on the faces of a magic object.
- 3. The VERTICES TUPLE is a tuple of numbers around the vertices of a three dimensional object (vertices tuple can be of 3 numbers in a square a tetrahedron and a dodecahedron, of 4 numbers in an octahedron and of 5 numbers in an icosahedron.
- 4. THE FACE ANGLES TUPLE is a tuple of numbers on the angles (or corners), of each face of a 3 dimensional object. That is a face angle tuple can be made up of 3 numbers in a tetrahedron and octahedron and icosahedron, 4 numbers in a square, and 5 numbers in a dodecahedron.
- 5. A POSITIVE MAGIC SQUARE is a magic square in which all the integers within the square are positive
- 6. A NON NEGATIVE MAGIC SQUARE is specific positive magic square where one of the integers equals 0 and all other integers are positive.
- 7. The MAGIC CONSTANT is a positive number to which all the rows and all the columns and the two diagonals in the magic square sum to.
- 8. EQUIVALENT MAGIC SQUARES are magic squares that can be derived one from the other by rotation or reflection.
- 9. The PIVOT is a positive integer located in the center of the square (exemplified in the number 5 in FIG. 1, and numbers situated in 150,250,350.450,550, and 650 in FIGS. 2a-i ).
- 10. OPPOSITE PAIR OF INTEGERS are the two integers on both sides of the pivot situated on a line through the pivot (exemplified in the numbers 3 and 7; and 9 and 1 in FIG. 1; and numbers situated in 130 and 170; 160 and 140; 120 and 180; 190 and 110 in FIG. 2b ).
- 11. a CUBE WITH MAGIC SQUARES is a cube comprised of 6 magic squares located on each of the cubes faces;
- 12. A CORNER TRIPLET is a triplet of integers located around the corner of the cube with magic squares. Each cube has 8 corner triplets (exemplified in the triplets 310,130,570; 330,410,590; 370,190,210; 390,230,470; 610,170,270; 630,490,290; 690,530,430 in FIGS. 2a-i );
- 13. An N TUPLE is a sequence of N numbers;
- 14. DIAGONALLY OPPOSITE SUBCORNER 4 tuple is a 4 tuple of integers consisting of two pairs of integers which are located below two diagonally opposite corners. There are 18 diagonally opposite subcorner 4 tuple4 tuples in a cube with magic squares. Exemplified as the four integers marked by a, the four integers marked by b, the four integers marked by c and the four integers marked by d in FIG. 3a , and numbers situated in 330, 410, 110 and 670 in FIGS. 3b 2a, 2d and 2f and numbers situated in 160,340,460 and 660; 130,310,430, and 190,370,490 and 630; in FIGS. 3b, 2a, 2e and 2 h;
- 15. DIAGONAL RING 6 TUPLE is a 6 tuple of integers located on the minor diagonals of three magic squares intersecting on a corner of the cube. Exemplified as the 6 integers marked by r in FIG. 3c . and numbers situated in 420,440,360,320,580,560; and 460,420,560,520,680,660 in FIG. 2a, 2h and FIG. 3 d;
- 16. MIDDLE HOOP 4 tuple is a 4 tuple including the 4 integers in the middle of the four rows surrounding the magic square on a face of the cube. There are 6 middle hoop 4 tuples in a square. Exemplified as the 4 integers marked by t, 4 integers marked by u, 4 integers marked by v, 4 integers marked by w, 4 integers marked by x, and the 4 integers marked by y, in FIGS. 3e and 4 tuple numbers situated in 580,440,220,160; 280,460,520,140; 680,120,320,420; 380,480, 620,180; 260,660,560, 360; 240,340, 540,640 in FIGS. 2a-h , and FIG. 3 f.

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, FIGS. 5 to 31 in general illustrate a three-dimensional object for use as an ornament or a puzzle. The object has generally a regular geometry and has multiple faces, and each face has numbers arranged at specific locations on the face. All faces may have the same layout of numbers. When used as a puzzle, some of the numbers can be removed or rotated away, while the remainder are fixed, and a preset rule links all of the numbers on the various faces, so that the remaining numbers plus the rule enable correct replacement of the numbers that were removed or rotated away to enable a correct solution of the puzzle.
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.
FIGS. 5 to 23 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 FIGS. 11 and 12 a-b, in which two or more different pivots may be applied to different ones of the six faces, say a first pivot to three of the faces and a second pivot to the remaining three faces. Each face may be a non-negative magic square with one or the other of the two pivots. The cube may have eight corner triplets, each summing to a respective constant which is related to one of the common pivots. The constant may be three times either one of the two pivots.
In one of the embodiments, as discussed in greater detail below in respect of FIGS. 13a-c , one row and one column is removed from each face, say the middle row and the middle column in each, to form a reduced two by two face. The reduced face cannot be a magic square but can be nonnegative, and may have four integers which are distinct. The four reduced faces may be distinct from one another and the respective diagonals of the reduced faces may sum to twice the common pivot. In addition the reduced faces may have eight corner triplets summing to a first constant, and/or eighteen diagonally opposite sub corner 4 tuples where the diagonally opposite sub corner 4 tuples and the faces together sum to a second constant.
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 FIGS. 24 to 26, the object may be a pyramid. The pyramid may have four triangular faces. Typically, each face has six positive numbers arranged in triangular form as three edges of three numbers with numbers at the vertices shared between different edges. Each triangle is non-equivalent, and the rule defines that all numbers in at least three faces are distinct. The fourth face, or base, may include three distinct pairs of equal numbers.
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 FIG. 24, a pair of the pyramids may share a common pivot number.
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 FIGS. 27 to 31, which show a star of David 270 comprising two interlinked oppositely facing triangles 272 and 274, each triangle having three vertices and three edges. The edges of the two triangles meet at six intersections 276, and a number is 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.
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 FIG. 4. He proved that every 3×3 positive (non negative) magic square fits this formula or can be derived through this formula:
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 FIG. 4, that positive (non negative) Magic Squares have the following properties:

- c a free parameter and a and b are bounded by c.
- c (the Pivot) is the average of all the integers in the square and is located in center cell of the square.
- The magic constant is equal to 3c.
- The maximal integer in the square is bounded by (equal or smaller than) 2c−1 for positive squares and by 2c for non negative squares
- Notice also that all magic squares with the same pivot (c) have the same magic constant and every pair of opposite integers in the magic square sum to 2c.

The Number of Non Equivalent Magic Squares.

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

- For positive magic squares where c is odd:

N=((c−1)/2)(c−3)/2))−a

- For positive magic squares where c is even:

N=(c/2−1)² −a

- For non negative magic squares where c is odd

N=(c−1)²/4−a

- For non negative magic squares where c is even:

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.

Wonder Cubes:

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:

- a. All the magic squares on the cubes faces are non-negative and are not equivalent to one another and have the same magic constant (3c).
- b. The 6 magic squares on its faces are linked as follows:
  - a. The eight corner triplets sum to the magic constant of the squares which is equal to 3c
  - b. The six middle hoop 4 tuples sum to one constant. In specific instances this constant equals 4c
  - c. The 18 diagonally opposite sub corner 4 tuples sum to one constant. In specific instances this constant equals 4c
  - d. The six opposite middle 4 tuples sum to one constant. In specific instances this constant equals to 4c
  - e. The 8 diagonal ring 6 tuples sum to sum to one constant. In specific instances this constant equals to 6c

Quasi Wonder Cubes,

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

- a. The constant of any magic square on the cube equals one of two constants, either 3c or 3d
- a. The eight corner triplets sum to one or two constants, in specific instances, these constants are either 3c and/or 3d

Derivatives of Wonder Objects

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:

- a. The six middle hoop 4-tuples that sum to 4c
- b. The six associated middle sub corner 4-tuples that sum to 4c
- c. The eight diagonal ring 6-tuples that sum to 6c
- d. The twelve arms of the six crosses on the faces of this body sum to 3c

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:

- a) the 8 corner triplets of the 2×2×3 cube sum to one constant. In specific instances it is equal to 3c
- b) the 18 diagonally opposite sub corner 4 tuples, and the faces of each cube, sum to one constant. In specific instances it is equal to 4c

Metamorphosis

In the following we show how to transform a wonder cube into two different bodies. FIG. 40 illustrates a cube with pivot 11. Various transformations of the cube with pivot 11 are illustrated in FIGS. 32 to 37, all based on the wonder cube with pivot equal to 11 but it can be performed on any wonder cube, of whatever pivot. In FIG. 40 a wonder cube whose pivot is equal to 11 is shown. We choose two corners at the two ends of a space diagonal, the diagonal connecting the bottom upper left corner to the top lower right corner of the cube. There are four possible choices of such space diagonals. Two such corners in FIG. 40 have the triplets (10,16,7) and (12,8,13) around them. The cube is now subdivided into two parts, as shown in FIGS. 34 and 35, each part including three faces of the cube that intersect at one end of the above chosen diagonal, respectively. The three faces of each of the two parts are now reset in parallelogram planar form, as shown in FIGS. 34b and 35 b.
Note that:

- The 6 parallelograms are in fact a copy of the six faces of the cube and are therefore magic squares with pivot equal to 11
- The triplets of parallelograms can be split into five layers: The first layer at the bottom of FIG. 34 include the integers 12,13 and 8 whose sum is 3×11=33.

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 FIG. 35 have the same properties. We have now the following two possibilities of transforming the two parallelogram triplets so as to get new Same Sum objects.

Platonic Dual Solids

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 FIGS. 32 and 33.
In greater detail, FIG. 32 shows a 2×2×3 wonder cube with a hidden pivot equal to 10. FIG. 33 represents an octahedron. The solid octahedron is constructed from the planar layout by joining the vertices marked by A, raising the joined A vertex up and pushing the O vertex down. Its 8 triangular faces are ABC, OBC, ACD, OCD, ABE, OBE, AED, and OED. The triangles are inscribed with three integers each and all the integers in FIG. 33 are the same integers as in FIG. 32, rearranged in a way such that there is a one to one correspondence between the 8 corner triplets that sum to 30 in the cube, and the triplets of numbers on the 8 faces of the octahedron. Moreover there is a one to one correspondence between the 6 4-tuples of integers around the 6 vertices of the octahedron that sum to 40, each, and the sets of 4 integers inscribed in the 6 faces of the cube. The puzzle solver is invited to find the correspondence between the triplets and the 4-tuples of both solids. In the example showed here the hidden pivot is equal to 10. The same duality can be constructed for 2×2×3 cube with any pivot, although in some cases the 3 integers inscribed in the triangular faces of the octahedron will not always be disjoint as they are in this example.

Double Tetrahedron Twins

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 FIG. 37. All we have to do is to join the two edges AB, the two edges AC, the two edges AD, the two edges DE and to bend the three parallelograms in FIG. 33 over the broken lines that split the parallelograms into two triangles, and do the same to the three joined parallelograms in FIG. 34. This creates two double tetrahedrons which inherit the properties of the parallelogram configurations so double tetrahedrons can be split into five layers summing to 3c, 6c, 9c, 6c and 3c. We refer to those two double tetrahedrons as TWIN, since there are links between them. Those links are shown in the illustration in FIG. 16 that lists 4-tuples of integers that sum to 4×c (=44 in this illustration) and are formed by two associated pairs of integers, one pair from each double tetrahedron. This is shown in FIG. 16 for the case where c=11. The three pairs of integers on both sides of the edge BE in FIG. 34, when joined to the associated corresponding pairs on both sides of the edge E′D′ in FIG. 35 result into 3 pairs of 4-tuples that sum to 4×11=44. We get a similar result when joining the three pairs of integers on both sides of the edge DE to the three pairs of integers on both sides of the edge E′C′ and we get the same result for the edges CE and E′B′. The two twin double tetrahedrons for the case where c=11 are shown in FIG. 37.
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 FIGS. 34 and 35, by joining the edges DED and D′E′D′ we get all the integers in the cube set upon a shield of David configuration as shown in FIG. 36. In this configuration the faces of the cube are transformed into six parallelograms joined together in a way such that the layer properties of the half cubes are preserved while each parallelogram is identical to the magic square on one of the faces of the cube.
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:

- 1. The formula is not unique. i.e. It is possible to find other formulas too for constructing stars of David with similar properties and this formula itself can provide several stars with the same magic constant depending on various choices of the four parameters a,b,c and d.
- 2. Any star of David whose creation is based on the above-mentioned formula has additional same sum properties: The integers on the three vertices of one triangle sum to the same constant as the three integers on the three vertices of the second triangle. The six small triangles around the shield can be separated into three pairs of triangles whose three numbers on their vertices sum to the same constant.
- 3. The parameters a, b, c, d, can be chosen in a way such that the integers on the shield have a national, cultural or personal meaning.
  Other planar derivatives of the wonder cubes are given in the examples below.

Implementation

- a. The 3×3×3 wonder cube can be designed as Rubik cubes and treated as the Rubik puzzle.
- b. The 3×3×3 and 2×2×3 wonder cubes with their magic properties can be designed as amulets or talismans or ornaments.
- c. The 2×2×3 cubes can be used in schools where exercises based on them can be designed so as to improve the arithmetical skills of the children.
- d. A new type of puzzles can be created, based on the 2×2×3 wonder cubes.
- e. Wonder cubes can be created in a way such that their various parameters (the pivot, the magic constant, the integers included in the same magic square etc.) have national, traditional, religious, or personal interpretations. Wonder cubes can be constructed to include important personal dates (birth days, marriage days etc.), or important dates in the national history as independence days, or important religious days or holiday dates. For example, mystical numbers in the Cabala, or other numbers of importance in the Jewish tradition: 5 (chamsa) for luck, 7 for the Sabbath or days of the week and creation of the world, 10 commandments, 12 tribes, 13—lucky number and age of Bar Mitzva etc. In particular the number 32 has an important number in the Cabala. Mystical numbers in the Christian tradition may include the number 33.
- f. 2×2×3 derived wonder cubes spread out in planar form can be set as a puzzle like the Sudoku puzzle or similar newspaper, journal puzzles, computer games, cellular phone based applications etc.

Specific Examples of Linked Magic Squares on a Cube (Wonder Cubes)

FIG. 5 shows an exemplary non negative wonder cube where the pivot is equal to 10 and the magic constant of the squares is 30. The wonder cube follows all required properties:

- a. The magic constant of all six squares equals 30
- b. The eight corner triplets sum to the magic constant 30
- c. the six middle hoop 4 tuples sum to 40
- d. the eighteen diagonally opposite sub corner 4 tuples sum to 40
- e. the six opposite middle 4 tuples sum to the constant 40
- f. the eight diagonal ring 6 tuples sum to the constant 60

FIG. 6 shows an exemplary positive wonder cube, with the pivot equal to 8. The wonder cube follows all required properties:

- a. The magic constant of all six squares equals 24
- b. The eight corner triplets sum to the magic constant 24
- c. the six middle hoop 4 tuples sum to 32
- d. the eighteen diagonally opposite sub corner 4 tuples sum to 32
- e. the six opposite middle 4 tuples sum to the constant 32
- f. the eight diagonal ring 6 tuples sum to the constant 48

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.
FIG. 7 shows an exemplary positive wonder cube with the pivot equal to 11. The magic constant of all its six squares is equal to 33.

- a. The magic constant of all six squares equals 33
- b. The eight corner triplets sum to 33
- c. the six middle hoop 4 tuples sum to 44
- d. the eighteen diagonally opposite sub corner 4 tuples sum to 44
- e. the six opposite middle 4 tuples sum to the constant 44
- f. the eight diagonal ring 6 tuples sum to the constant 66.

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)
FIG. 8 shows an exemplary positive wonder cube with the pivot equal to 12.

- a. The magic constant of all six squares equals 36
- b. The eight corner triplets sum to the magic constant 36
- c. the six middle hoop 4 tuples sum to 48
- d. the eighteen diagonally opposite sub corner 4 tuples sum to 48
- e. the six opposite middle 4 tuples sum to the constant 48
- f. the eight diagonal ring 6 tuples sum to the constant 72

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.
FIG. 9 shows an exemplary positive wonder cube with the pivot equal to 16.

- a. The magic constant of all six squares equals 48
- b. The eight corner triplets sum to 48
- c. the six middle hoop 4 tuples sum to 64
- d. the eighteen diagonally opposite sub corner 4 tuples sum to 64
- e. the six opposite middle 4 tuples sum to the constant 64
- f. the eight diagonal ring 6 tuples sum to the constant 96.

FIG. 10 shows an exemplary positive wonder cube with the pivot equal to 18.

- a. The magic constant of all six squares equals 54
- b. The eight corner triplets sum to 54
- c. the six middle hoop 4 tuples sum to 72
- d. the eighteen diagonally opposite sub corner 4 tuples sum to 72
- e. the six opposite middle 4 tuples sum to the constant 72
- f. the eight diagonal ring 6 tuples sum to the constant 108.

FIG. 11 shows an exemplary quasi wonder cube with the pivots equal to 5 and 8.

- a. The constant of any magic square on the cube equals either 15 or 24
- b. The eight corner triplets split into 2 sets, one set which sums to 15 and the other set sums to 24
  The earliest mention of a magic square can be found in Chinese literature, dating as early as 650 BC, where a legend about a flood of the river LO calmed when the emergence from the river of a turtle, with a 3×3 magic square on its shell (the LO SHU magic square shown in FIG. 1), with magic constant equal to 15, telling the people how many sacrifices they have to make in order to calm the river. The number 15 happens also to be the number of days in each of the 24 cycles of the Chinese solar year. One of the squares on this cube is identical to the LO SHU magic square.

FIG. 12a shows an exemplary quasi wonder cube with the pivots equal to 8 and 12.

- a. The constant of any magic square on the cube equals either 24 or 36
- b. The eight corner triplets all sum to 32.
- This cube has Jewish implications. 32 is an important number according to the Cabala. It is the sum of 22, the number of letters in the Hebrew alphabet, and 10, the number of digits. 32, written as letters using Gematria (a letter-number equivalence system in Jewish tradition) is LEV which means “heart” in Hebrew. Finally 32 is 2 to the power of five which relates to the traditional Jewish belief that our world is five dimensional, the three geometric dimensions, the time dimension and the “good and bad” dimension, with all five dimensions infinite in both directions. 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.

FIG. 12b shows how an exemplary 2×2×3 cube is derived from a 3×3×3 wondercube. A full wondercube is taken and all the middle rows and columns are deleted—as indicated by the squares in the figure that are blanked out.
FIGS. 13a-c shows an exemplary 2×2×3 cube derived from a wonder cube with the pivot equal to 10 (shown in FIG. 5).
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.
FIG. 14 shows an exemplary 2×2×3 cube derived from a wonder cube with the pivot equal to 8 (shown in FIG. 6).
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.
FIG. 15 shows an exemplary 2×2×3 cube derived from a wonder cube with the pivot equal to 11(shown in FIG. 7).
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.
FIG. 16 shows an exemplary 2×2×3 cube derived from a wonder cube with the pivots equal to 12 (shown in FIG. 8).
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.
FIG. 17 shows an exemplary 2×2×3 cube derived from a wonder cube with the pivot equal to 16 (shown in FIG. 9).
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
FIG. 18 shows an exemplary 2×2×3 cube derived from a wonder cube with the pivots equal to 18 (shown in FIG. 10).
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.
FIG. 19 shows an exemplary 2×2×3 cube derived from a quasi wonder cube with the pivots equal to 5 and 8 (shown in FIG. 11).
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
FIG. 20 shows an exemplary 2×2×3 cube derived from a quasi wonder cube with the pivots equal to 8 and 12 (shown in FIGS. 12a-b ).
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.
FIG. 21a is an example of a puzzle game based on the exemplary cube with c=11. It is designed in such a way that all the 24 corner cells (3 in each of the 8 corners) can be removed from the cube and reset on it (in different locations).
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.
FIGS. 21b-c is an example of a puzzle game based on the exemplary cube with p=11. Remove all corner cells from the cube. What is left is a cube with cross shaped objects on the 6 faces of the cube. The cube is designed in a way such that the cross shaped objects can be removed and reset on the cube. The cross shaped objects have the same 5 integers inscribed on both sides such that one side is the reflection of the other. REMOVE one to five of the cross shaped objects and turn some around to show the side, then scramble them then reset them on the cube so as to restore the original cube with linked properties between the faces. The more cross shaped objects removed the more difficult the challenge, again at the player's discretion. Notice that some of the properties of the original cube are inherited by the cube with cross shaped objects, specifically, the six opposite middle 4 tuples sum to one constant. In specific instances this constant equals 4c, where c is the pivot.
FIG. 22a shows a derived 2×2×3 derived from a wonder cube with P=12
FIG. 22b shows a derived 2×2×3 from wonder cube with p=8.
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.
FIGS. 23a and 23b show the planar spread out of an exemplary cube derived from a wonder cube with p=8. The game is similar. Parts of the puzzle can be removed and reset to form the linked squares and to this end the corners are marked by the letters A to H.
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 FIG. 24, which shows two magic pyramids 240 and 242 according to embodiments of the present invention. Pyramids 240 and 242 share the following properties: The sum of the 3 integers along any of the edges of the pyramids is equal to 33. The triplets of integers around any of their corners also sum to 33. The two pyramids have also a linked property.
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.

Twin Magic Pyramids

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.
FIG. 25 shows a planar layout of a magic pyramid derived from a square with a pivot of C=11.
The pyramid has the usual magic properties i.e. every one of its 4 faces includes

- a) 6 positive integers arranged in triangular form.
- b) The triangles of integers on the faces of the pyramid are not equivalent to one another.
- c) all integers in any of the faces are distinct except for the bottom face which includes 3 distinct pairs of equal integers.
- d) the 3 integers along any of the sides of the triangle sum to the same magic constant which is equal to 33.
- e) The pivot number (in this case 11) is found in each face, in the middle of the three tuple adjacent to the bottom edge.
- f) the bottom face is comprised of 3 pairs of distinct integers.
- g) The linked property of the pyramid: every 3 integers around every one of it's 4 vertices sum to a specific number. In specific instances this number equals (3c)
- h) The belt comprised of 6 integers in the middle part of the pyramid (when U1 is combined with U2 and U3) namely 9,15,10,20,4,8 sum to a specific number. In specific cases this number equals 6c. in this case 66=6*11.

FIG. 26 shows a planar layout of the twin pyramid 242 derived from the square with pivot c=11, which is comprised of faces with distinct integers which are not equivalent to one another or to the faces of the first twin pyramid 240. The twin pyramid 242 has the same properties as described above.
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 FIG. 26 merge into an edge (U,Q) on the 3 dimensional pyramid. On both sides of this edge there are 3 pairs of corresponding integers: (10,7), (15,10) and (8,16). In a similar way, on the twin pyramid, there is an edge (A,B) on the pyramid shown schematically in FIG. 25, which has on both its sides the three pairs of corresponding integers (20,7),(1,18) and (12,8).
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:

- i) The sum of the 4 tuple comprised of the pair in the first pyramid and the corresponding pair in the second pyramid equals a specific number. In a specific case this number equals 4c (in this case 44=4*11). In this case, there are eight such 4 tuples comprised of corresponding pairs, which will add to 4c.
  Games that can be associated with this, or similar, twin pyramids:
- 1) In one pyramid: to remove some of the integers, scramble them then restore the pyramids to their original form
- 2) In two twin pyramids: to find all twin edges, in the above sense, that exist in the 2 twin pyramids.

Reference is now made to FIGS. 27 to 31, which disclose specific magic stars of David.
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)
FIGS. 27-31 shows exemplary Stars of David with integers on the corners of the triangles and at their intersections. Adding the 4 integers along any of the 6 sides of the triangles will sum to the same magic constant.
The magic constant in FIG. 27 is 47 star 270 also includes the integers 20,9;11,19 hinting to the date Nov. 29, 1947 (47 being the magic constant) and this the date when the U.N accepted a resolution to establish the state of Israel. The magic constant in star 280 in FIG. 28 is 48 and the integers included in this star contain the integers 15,5,19, hinting at the date May 15, 1948 when Israel was created. When placing the stars of FIGS. 27 and 28 together, one alongside the other, the 6 numbers in the form of a hexagon created by the right side of the left star with the left side of the right star sum to 70, pointing to the 70th anniversary of the state of Israel.
Reference is now made to FIG. 38, which illustrates four platonic solids, a) a tetrahedron, b) a cube, c) an octahedron made of a double pyramid, d) a dodecahedron, and e) an icosahedron. In the top row are shown the pure shapes and in the bottom row how numbers may be distributed over the shape according to an embodiment of the present invention.
Reference is now made to FIG. 39, which illustrates upper and lower parts of a dodecahedron. The parts are shown in planar form and may be cut out and folded to fit together to form the final shape.
Reference is now made to FIG. 40 illustrates a cube with pivot 11 and was discussed hereinabove in respect of the transformations of FIGS. 32 to 37.
Reference is now made to FIG. 41, which illustrates upper and lower parts of an icosahedron. The parts are shown in planar form and may be cut out and folded to fit together to form the final shape.
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.

Claims

What is claimed is:

1. 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.

2. The three-dimensional object of claim 1, wherein said locations comprise a center location, edge locations and corner locations and numbers at each location.

3. The three-dimensional object of claim 1, wherein said removable numbers are mounted on buttons, said buttons being one member of the group consisting of magnetic buttons, clip-on buttons, and buttons mounted on rotatable levers.

4. The three-dimensional object of claim 1, being one member of the group consisting of a cube, a pyramid having four triangular faces, a tetrahedron a double pyramid, a double tetrahedron, an octahedron, a dodecahedron, and an icosahedron.

5. The three-dimensional object of claim 4, wherein each face has nine numbers arranged in three rows of three, or wherein each face is a magic square having a pivot number, and said rule defines a common pivot number for respective faces.

6. The three-dimensional object of claim 4, wherein each face is a magic square having a pivot number, and said rule defines a common pivot number for respective faces, the object further having eight corner triplets, each one of said eight corner triplets summing to three times said common pivot number.

7. The three-dimensional object of claim 1, having a further rule being one member of the group consisting of:

having six middle hoop 4 tuples, each of said six middle hoop 4 tuples summing to a common constant,

having eighteen diagonally opposite sub corner 4 tuples, said diagonally oppositely located sub corner 4 tuples summing to one constant, and

having six opposite middle 4 tuples, said six opposite middle 4 tuples summing to a common constant,

having a common constant being four times said common pivot,

having eight diagonal ring six tuples, each of said eight diagonal ring six tuples summing to a second common constant, and

having a second common constant being six times said common pivot.

8. The three-dimensional object of claim 5, wherein 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, or wherein said respective constant is three times said related one of said common pivots.

9. The three-dimensional object of claim 5, being a cube having faces, each face having nine numbers arranged as three rows in three columns, all faces having a common pivot, 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, 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, or wherein the first constant is three times the common pivot and/or the second constant is four times the common pivot.

10. The three-dimensional object of claim 4, being the pyramid having four triangular faces, wherein 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.

11. The three-dimensional object of claim 10, wherein said rule defines that one of said faces includes three distinct pairs of equal numbers, or wherein all the edges of said pyramid sum to a first common constant, or wherein said first common constant is three times a pivot number, said pivot number being present in each face, or wherein the numbers of the three middle rows of said first three faces extending as a belt around said pyramid sum to a second constant, or wherein said second constant is six times a pivot number, said pivot number being present in each face.

12. A pair of the pyramids of claim 10, said pyramids sharing a common pivot number, or wherein said pair of pyramids are 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, or wherein said common constant is four times said common pivot number.

13. A three-dimensional object, 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.

14. The three-dimensional object of claim 13, being one member of the group consisting of a cube, a pyramid, a tetrahedron a double pyramid, a double tetrahedron, an octahedron, a dodecahedron, and an icosahedron and wherein said predetermined rule links all of said numbers on the plurality of faces to enable correct replacement of said removed numbers.

15. The three-dimensional object of claim 13, 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.

16. 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.

17. The object of claim 16, having additional numbers on each of the faces, and

wherein an additional predetermined rules link all of said numbers on the plurality of faces.

18. The object of claim 16, being one member of the group consisting of a tetrahedron, a cube a double pyramid, a double tetrahedron, an octahedron, a dodecahedron, and an icosahedron.

19. The object of claim 16 in a planar representation, and/or wherein at least one of the numbers is removed from a respective location on at least one of the faces of the object.

20. The object of claim 19, wherein said removing of at least one of the numbers is to obtain a derivative object, said derivative object retaining at least said rule, or wherein said removal of at least one of the numbers is to allow replacement of said at least one of the numbers so as to keep at least said rule.