# Patent application title: DIVIDING METHOD FOR THREE-DIMENSIONAL LOGICAL PUZZLES

##
Inventors:
Maxime Paquette (Val-Des-Monts, CA)

IPC8 Class: AA63F908FI

USPC Class:
273153 S

Class name: Amusement devices: games puzzles shifting movement

Publication date: 2009-04-09

Patent application number: 20090091080

## Abstract:

A dividing method used to easily divide any given solid into perfectly
interfitting parts by using at least one guiding polyhedron to establish
an axis system serving as guiding paths for associated geometrical figure
contours used to slice said given solid. This axis system is coincident
with all or a subset of the geometrical centers of each face of the
guiding polyhedron, with midpoints of the edges of the polyhedron, and
with the vertices of the polyhedron. The dividing method is based on five
different techniques: a selecting technique, a sizing technique, a
multi-slicing technique, a multi-pivoting technique, and a multi-guiding
technique. This dividing method can create extremely challenging,
aesthetic and symmetrical three-dimensional puzzles having shifting and
optionally sliding features. This dividing method works with
polyhedron-based solids, spherical solids and odd-shaped solids of any
kind.## Claims:

**1.**A method of dividing any given solid into perfectly interfitting parts covering an entire outer surface of a shiftable three-dimensional puzzle, the method comprising steps of:selecting at least one guiding polyhedron;defining an axis system based on the at least one guiding polyhedron, wherein axes of the axis system passthrough all or a subset of geometrical centers of the faces, edges and vertices of the guiding polyhedron;associating, with each axis, a planar geometrical figure contour which can be projected along each respective axis into an intersection with the given solid to be divided; anddividing the given solid using the geometrical figure contour into perfectly interfitting parts covering the entire outer surface of the puzzle.

**2.**The dividing method as claimed in claim 1 wherein the step of associating the geometrical figure contour with each axis comprises steps of selecting a proper form for each geometrical figure contour associated with the axes of the axis system and sizing each geometrical figure contour for dividing the given solid.

**3.**The dividing method as claimed in claim 2 further comprising a step of applying a multi-slicing technique wherein said given solid is sliced more than once along one or more of the axes of the axis system with geometrical figure contours of a different size.

**4.**The dividing method as claimed in claim 2 further comprising a step of applying a multi-pivoting technique wherein a circular geometrical figure contour is added to one or more axes of the axis system to divide said given solid into pivoting groups of one or more elements.

**5.**The dividing method as claimed in claim 2 further comprising a step of applying a multi-guiding technique wherein one or more guiding polyhedra are superimposed as guides for multiple axis systems, with axes passing through all or a subset of geometrical centers of faces, edges and vertices of the guiding polyhedral whereby each axis of every additional axis system is associated with a geometrical figure contour which can be projected into an intersecting relationship with the solid in order to slice the given solid into perfectly interfitting parts covering the entire outer surface of the solid.

**6.**The dividing method as claimed in claim 1 comprising at least one of the steps of:selecting a proper form for each geometrical figure contour associated with axes of the axis system;sizing each geometrical figure contour to be used for slicing the given solid;applying a a multi-slicing technique wherein the given solid is sliced more than once along at least one axis of the axis system with a geometrical figure contour of a different size;applying a multi-pivoting technique wherein a circular geometrical figure contour is added to at least one axis of the axis system to divide said given solid into pivoting group of one or more elements; andapplying a multi-guiding technique wherein one or more guiding polyhedra are superimposed as guides for axis systems, with axes passing through all or a subset of geometrical centers of faces, edges and vertices of the guiding polyhedra, whereby each axis of each additional axis system is associated with a geometrical figure contour along which the geometrical figure contour can be projected into an intersecting relationship with the solid in order to slice the given solid into perfectly interfitting parts covering the entire outer surface of the solid.

**7.**The dividing method as claimed in claim 6 wherein the guiding polyhedra are convex uniform polyhedra selected from the five platonic solids, the thirteen archimedean solids, the prism solids, and the antiprism solids.

**8.**The dividing method as claimed in claim 7 wherein most of the associated geometrical figure contours are circular in order to create a mostly symmetrical three-dimensional puzzle when said given solid is divided, wherein some of the interfitting parts act as pivoting elements while enabling substantially all of the other parts of the puzzle to be shifted.

**9.**The dividing method as claimed in claim 8 wherein said given solid is a polyhedron.

**10.**The dividing method as claimed in claim 9 comprising a further step of superimposing sliding elements onto one or more outer surfaces of said puzzle.

**11.**The dividing method as claimed in claim 8 wherein said given solid is a sphere.

**12.**The dividing method as claimed in claim 11 comprising a further step of superimposing sliding elements onto one or more outer surfaces of said puzzle.

**13.**The dividing method as claimed in claim 8 wherein said given solid is an odd-shaped solid.

**14.**The dividing method as claimed in claim 13 comprising a further step of superimposing sliding elements onto one or more outer surfaces of said puzzle.

## Description:

**TECHNICAL FIELD**

**[0001]**The present invention relates generally to a dividing method useful for simply dividing any given solid into perfectly interfitting parts, or making three-dimensional logical puzzles and, in particular, to puzzles having either a spherical shape or a shape based on a polyhedron.

**BACKGROUND OF THE INVENTION**

**[0002]**The prior art of shifting-movement puzzles includes regular, semiregular and irregular polyhedra. There are numerous types of polyhedron-based puzzles known in the art. Most of the prior art polyhedron puzzles are based on the five platonic solids and are of very moderate complexity.

**[0003]**Also known in the art are three-dimensional sliding puzzles. Three-dimensional puzzles combining shifting and sliding features have been proposed by Applicant in U.S. patent application Ser. No. 11/738,673 (Paquette) entitled "Three-Dimensional Logical Puzzles", which was filed on May 2, 2007.

**[0004]**Also known in the art are ball-shaped or spherical puzzles. Spherical shifting puzzles are very scarce due to the great difficulty of properly dividing a sphere in order to obtain a symmetrical, aesthetical and challenging puzzle.

**[0005]**Spherical puzzles created by dividing a sphere based on a guiding regular polyhedron, i.e. by defining outer spherical sections by dividing the sphere parallel to a guiding polyhedron to create overlapping spherical sections on the sphere, are proposed by Applicant in U.S. patent application Ser. No. 11/738,673, supra. A spherical puzzle created by this technique is challenging, entertaining and aesthetically pleasing.

**[0006]**Odd-shaped puzzles, such as a human head for example, are proposed but are of a low difficulty level again due to the complexity of the shape division involved.

**[0007]**Therefore, complexly subdivided regular, semiregular or irregular polyhedron-based puzzles, or spherical puzzles, or odd-shaped puzzles enabling shifting (and optionally also sliding movement) would provide a highly challenging, entertaining and aesthetically-pleasing three-dimensional puzzle.

**SUMMARY OF THE INVENTION**

**[0008]**An object of the present invention is to provide an easy, straightforward dividing method useful for making symmetrical, challenging, entertaining and aesthetically pleasing polyhedron-based, or spherical-based, or odd-shape-based puzzles having elements that can be shifted and which can optionally further include superimposed sliding features.

**[0009]**The present disclosure explains a method of dividing any given solid in perfectly interfitting parts by using an axis system associated with a guiding polyhedron. The axes are defined as passing through all or a subset of the geometrical centers of every face, edge midpoints and vertices. Each axis serves as a path along which a planar (two-dimensional) geometrical figure can be projected into an intersecting relationship with the given solid to thereby slice the given solid into perfectly interfitting parts according to the particular contours of the geometrical figure. In other words, a plurality of potentially different geometrical figures, each defining a cutting plane having its own geometrical contours, is used to cuts, or slice, the solid into puzzle elements by intersecting the solid with the various geometrical figures whose respective orientations remain fixed relative to their respective axes.

**[0010]**By properly choosing a suitable guiding polyhedron, axis system and associated geometrical figures, an infinity of aesthetic and challenging three-dimensional puzzles can be produced from various solids.

**[0011]**The exposed dividing method works with polyhedron solids, spheres and odd-shaped solids of any kind. Any polyhedron can be selected as the guiding polyhedron, but the preferred ones for symmetrical reasons are of the convex uniform kind, such as the platonic solids, the archimedean solids and the prism and antiprism solids.

**[0012]**The dividing method exposed in the present disclosure can be easily extended by using superposed polyhedra for guiding purposes, all of which lies within the scope of the present invention.

**BRIEF DESCRIPTION OF THE DRAWINGS**

**[0013]**The embodiments of the present invention will now be described with reference to the appended drawings in which:

**[0014]**FIG. 1 is a schematic representation of the proposed dividing method;

**[0015]**FIG. 2 illustrates a sphere divided in rotating, mobile and gap elements by circular geometrical figures associated with a guiding tetrahedron;

**[0016]**FIG. 3 is showing a geometrical figure circular radius r

_{ng}selected to eliminate the gap elements (tetrahedron face guided);

**[0017]**FIG. 4 is showing a sphere divided by a circular geometrical figure with a circular radius smaller than the no-gap radius r

_{ng}(tetrahedron face guided);

**[0018]**FIG. 5 presents the outcome of a dividing radius superior to the no-gap radius r

_{ng}(tetrahedron face guided) illustrating the sizing technique of the dividing method;

**[0019]**FIG. 6 shows a double slicing at radius r

_{2}ng and r

_{max}of the sphere from FIG. 5 representing the multi-slicing technique aspect of the dividing method (tetrahedron face guided);

**[0020]**FIG. 7 also illustrates a double-slicing of a sphere from FIG. 5 with a second radius non-equal to the no-gap radius r

_{2}ng (tetrahedron face guided);

**[0021]**FIG. 8 illustrates a combination of a double-sliced circular tetrahedron face division with a circular tetrahedron vertex division presenting the multi-pivoting technique of the dividing method;

**[0022]**FIG. 9 illustrates a circular dodecahedron face division of a sphere;

**[0023]**FIG. 10 is a circular dodecahedron face division of a sphere at the no-gap radius r

_{dng};

**[0024]**FIG. 11 illustrates the outcome of a dividing radius superior to the no-gap radius r

_{dng}(dodecahedron face guided);

**[0025]**FIG. 12 illustrates a double-sliced circular dodecahedron face division of a sphere;

**[0026]**FIG. 13 is showing a circular icosahedron face division of a sphere;

**[0027]**FIG. 14 shows a combination of a single-sliced circular icosahedron face division with a multi-pivoting single-sliced circular icosahedron vertex division of a sphere;

**[0028]**FIG. 15 is a schematic representation of the dividing method applied to a cube using an octahedron guiding polyhedron;

**[0029]**FIG. 16 illustrates a single-sliced six-pointed star octahedron face to cube vertex division, with a single-sliced octahedral octahedron vertex to cube face division, and a single-sliced hexahedral octahedron edge to cube edge division of a cube for non-puzzle purposes;

**[0030]**FIG. 17 illustrates a single-sliced circular octahedron face to cube vertex division of a cube suited for puzzle purposes;

**[0031]**FIG. 18 is a combination of a single-sliced circular octahedron face to cube vertex division with a mutli-pivoting single-sliced circular octahedron vertex to cube face division applied to a cube;

**[0032]**FIG. 19 illustrates an exploded view of an odd-shape puzzle divided by a single-sliced tetrahedron face division;

**[0033]**FIG. 20 is an assembled view of FIG. 19 showing the extended possibilities of the dividing method applicable to any odd-shaped solid.

**[0034]**These drawings are not necessarily to scale, and therefore component proportions should not be inferred therefrom.

**DESCRIPTION OF THE PREFERRED EMBODIMENTS**

**[0035]**By way of introduction, the dividing method will be illustrated with simple preferred embodiments related to a regular guiding polyhedron. It is to be understood that any polyhedron or combination of polyhedra can be used as the guiding polyhedron associated with said axis system, all within the scope of the present invention.

**[0036]**The dividing method presented in this disclosure consists of a combination of techniques, a selecting technique, a sizing technique, a multi-slicing technique, a multi-pivoting technique, and a mutli-guiding technique used to create symmetrical, aesthetic and challenging puzzles, or simply used to divide any given solid into perfectly interfitting parts. Of the five techniques presented herein only the two first are essential to the dividing method. The three remaining techniques are optionally used to enhance the puzzle's complexity in order to achieve a greater challenge.

**[0037]**Reference is now made to FIG. 1. In this schematic representation of the dividing method, the given solid to be divided is represented by a sphere S with an inscribed tetrahedron T. The guiding tetrahedron could have been shown inscribed or subscribed without any differences since it is the axis system associated with the polyhedron that is relevant to the dividing method. For each face of the guiding polyhedron there is an associated axis f-f also associated with a geometrical figure F. Each axis f-f, four in the case of a tetrahedron, can be associated with a different geometrical figure F, or share the same one. The axis system is completed by having vertex axis v-v and midpoint edge axis e-e associated respectively with geometrical figures V and E. The preceding selection of the guiding polyhedron, axis system, and associated geometrical figures is said to be the selecting technique of the dividing method. It is the contour of the associated geometrical figures that will be used through projection along their respective axes to divide, or slice, the given solid into perfectly interfitting parts.

**[0038]**As mentioned the first technique involved in the dividing method is the selecting technique. This technique refers mainly to the selection of the proper form of every geometrical figure associated with an appropriate axis system to be used for slicing the given solid. The second technique involved in the dividing method is the sizing technique. This technique refers to the selection of the proper dimension, or size, of every associated geometrical figure. Proper selection of the geometrical figures and proper sizing of these figures are essential to the dividing method and depend on the expected purposes of the divided solid. As a general rule for puzzle purposes, very symmetrical parts are sought and as many as possible parts should be interchangeable (shifting-wise). So mostly circular figures are used for puzzle purposes with quite a bit of overlapping of the geometrical figures.

**[0039]**Reference is now made to FIG. 2 illustrating a sphere divided by a circular geometrical figure of radius r associated with the face axis f-f of a guiding tetrahedron T. The sphere is divided into four rotating elements 21, six mobile elements 22 and four gap elements 23.

**[0040]**The mobile elements 22 are grouped around each of the rotating elements 21 in shifting sections whereby mobile elements of one group can be interchanged with mobile elements of other groups. Thus, a shifting spherical puzzle is created by dissecting a sphere with cutting circular geometrical figures that are associated with each face axis f-f of a guiding tetrahedron T to generate overlapping outer spherical sections, each centered about a respective rotating element 21.

**[0041]**Necessary adjustments to convert the given solid elements into a functioning puzzle are well described in the prior art and need no further explanation other than mentioning that:

**[0042]**(i) each rotating element is connected to the puzzle by a retaining means, i.e. a fastener, fastener subassembly, retainer or other retaining mechanisms. These retaining means hold the pieces in an interfitting relationship and enable rotational movement around the associated axis. These retaining means could include a coil spring to reduce friction generated between adjoining surfaces and provide easily movable elements that are not prone to jamming, catching or getting "hung up". These interconnecting means could be replaced by snapping-action parts, which would also fall within the scope of the present invention;

**[0043]**(ii) holding means are provided for holding the remaining elements in an interfitting relationship with each respective rotating element, or adjacent remaining elements. Usually, the angles formed in the divided parts are such that remaining elements cannot slide out of their fitted position, thus preventing disassembly of the puzzle. Other interfittings, mechanisms or locking means are possible that enable elements to be interchanged from one group or subgroup to another group or subgroup by "shifting" (i.e. twisting or rotating) one group or subgroup relative to the other groups or subgroups. For example, locking means could include a tongue and groove mechanism. It is understood that this groove could be male (protrusion) or female (cavity), and of many shapes like dovetail-shaped, L-shaped or T-shaped or of any shape that provides a retaining means allowing rotation about an axis, all within the scope of the present invention;

**[0044]**(iii) the obtained puzzle can be designed with or without a center element or core located inside of the given solid puzzle, which can be either (a) an inner sphere, or (b) an internal concentric polyhedron, or (c) an axial rod (pivot) system. Depending on the guiding polyhedron used and the selected dividing geometrical figures, the center element may or may not have exposed faces. A coreless puzzle can be constructed by providing the rotating elements, mobile elements, gap elements, and the remaining elements, if applicable, with appropriate protrusions and grooves. These protrusions and grooves cooperate as interfitting male and female connections to slideably and rotatably interlock the various elements to thus hold the elements together to form a complete solid puzzle. Also, the center element could be constructed by the interfitting or snapping action of two half center core elements. When assembled together these two half center core elements form a hollow center core element shaped as a polyhedron or a sphere. With this hollow center core element, the rotating elements are rotationally connected to the core element by a screw inserted from inside the puzzle and thus no capping of elements is required in order to obtain an even and smooth outer surface over the given solid outer shell of the puzzle. All of the previously mentioned possibilities or modifications lie within the scope of the present invention.

**[0045]**The foregoing adjustments (or other similar adjustments well within the capabilities of a person of ordinary skill in the art) are needed to convert the given solid elements in the puzzles presented in the remaining figures of this disclosure so as to obtain functioning (shiftable) puzzles. These modifications and adjustemnts are well within the reach of a person familiar with the art of three-dimensional puzzles and therefore require no further elaboration.

**[0046]**FIG. 3 to FIG. 5 illustrate the huge influence that the sizing technique has on the resulting puzzle parts. That influence is translated into different types, forms and numbers of elements.

**[0047]**Reference is now made to FIG. 3 which shows a spherical solid divided by circular geometrical figures of radius r

_{ng}associated with the face axis of a guiding tetrahedron. It is the exact same selecting technique of the dividing method used in FIG. 2 except that the sizing of the circular geometrical figures is selected to eliminate the gap elements 23 of FIG. 2. This is achieved when the slicing contours meet at point P. Thus the obtained puzzle only has two types of elements, rotating elements 31 and mobile elements 32.

**[0048]**Reference is now made to FIG. 4 showing the result of the exact same selecting technique of the dividing method used in FIG. 2 and FIG. 3 with a circular radius r

_{u}of the geometrical figures sized smaller than the no-gap radius r

_{ng}. This division of the given solid is said to be "tetrahedron face guided" exactly as with the previous puzzle. This puzzle is constituted of three types of elements, pivoting elements 41, mobile elements 42, and gap elements 43.

**[0049]**Reference is now made to FIG. 5 illustrating a tetrahedron face guided division of a sphere with a sizing of the circular geometrical figure radius r

_{max}superior to the no-gap radius r

_{ng}. This puzzle is also constituted of three types of elements, pivoting elements 51, mobile elements 52, and gap elements 53 of different size and shape compared to the puzzle depicted in FIG. 4.

**[0050]**FIG. 6 and FIG. 7 illustrate the multi-slicing technique used to increase the number of parts that are interchangeable on a given solid puzzle. This multi-slicing technique is simple, straightforward and very efficient in creating challenging symmetrical puzzles.

**[0051]**Reference is now made to FIG. 6 where the multi-slicing technique is presented in the form of a double slicing at radius r

_{max}and r

_{2}ng (contours meeting at point P) of the sphere from FIG. 5. The spherical puzzle is now constituted of five different types of elements, rotating elements 61, mobile elements 62, secondary mobile elements 63, secondary gap elements 64, and inter-gap elements 65. Since the multi-slicing technique of a given solid puzzle using the same proposed selecting and sizing techniques will easily result in a different quantity of elements and different types of elements, it allows one to vary the total number of puzzle puzzle elements to achieve either simpler or more complex puzzles. It is to be understood that these simpler or more complex puzzles are within the scope of the invention presented in this disclosure. Also to be understood is that various combinations, changes or modifications are possible giving almost an infinity of possibilities if the dividing method is used with other regular, semiregular, irregular, spherical or odd-shaped given solid.

**[0052]**Reference is now made to FIG. 7 illustrating a tetrahedron face guided double-slicing of the sphere from FIG. 5 with a first radius r

_{1}and a second radius r

_{2}different from the no-gap radius r

_{2}ng. The outcome of this division is now six different types of elements, rotating elements 71, mobile elements 72, secondary mobile elements 73, secondary gap elements 74, inter-gap elements 75, and gap elements 76.

**[0053]**FIG. 8 presents another technique being part of the dividing method, the multi-pivoting technique. This technique is used to incorporate pivoting features around previously non-pivoting elements. The multi-pivoting technique is also very simple, straightforward and also very efficient for increasing the number of parts interchangeable on a given solid puzzle, thus creating ultimately challenging symmetrical puzzles.

**[0054]**Reference is now made to FIG. 8 illustrating the puzzle of FIG. 7 submitted to the multi-pivoting technique by introducing a said "tetrahedron vertex division" through circular geometrical figures of radius r

_{3}. This tetrahedron vertex division is carried along axes coincident with the guiding polyhedron vertices. It can be appreciated that the complexity of the puzzle rapidly increases. The puzzle being now constituted of ten elements, rotating elements 81, mobile elements 82, secondary mobile elements 83, secondary gap elements 84, inter-gap elements 85, gap elements 86, secondary pivoting elements 87, first inter-mobile elements 88A, second inter-mobile elements 88B, and tertiary mobile elements 89.

**[0055]**The last technique of the dividing method, the "multi-guiding technique", relates to the use of multiple guiding polyhedra used to divide one given solid. This technique corresponds to the superposition of different divisions from different puzzles into only one puzzle. The results of such superposition becomes rapidly complex and for the sake of simplicity only puzzles based on single guiding polyhedra are presented in this disclosure. However, it will be obvious to a person familiar with the art of three-dimensional puzzles, that this technique alone is an extremely powerful tool to create astonishingly complex and intriguing puzzles aimed at the expert enthusiast. But as mentioned in the prior art, with proper indicia pattern selection, the puzzle difficulty level can be modulated to obtain a reasonably solvable puzzle. FIG. 9 to FIG. 18 will now be devoted to illustrate the power of changing the guiding polyhedron used to divide a simple solid. This will enable a puzzle developer to appreciate the powerful dividing method disclosed herein. It is to be mentioned that only some of the five platonic solids are used as the guiding polyhedron in the present disclosure, but that any polyhedron showing some kind of symmetry can be used for puzzle purposes. This requirement is not necessary if only a division of a given solid into perfectly interfitting parts is sought. Accordingly, the best suited polyhedra to be used in the dividing method for puzzle purposes are of the convex uniform kind, such as the five platonic solids, the thirteen archimedean solids, and mostly all of the prism and antiprism solids.

**[0056]**Reference is now made to FIG. 9 illustrating a circular dodecahedron face division of a sphere. The associated geometrical figures are circular with a contour radius r

_{d}. The result of this division is twelve pivoting elements 91, thirty mobile elements 92, and twenty gap elements 93. This exact puzzle is presented in the prior art with a different dividing method. The dividing method of the present disclosure is more general.

**[0057]**Reference is now made to FIG. 10 showing a circular dodecahedron face division of a sphere at the no-gap radius r

_{dng}. With such a radius the contours projected on the sphere surface meet at point P, and thus no gap elements are produced leaving only pivoting elements 101 and mobile elements 102 covering the entire outer surface of the divided sphere. One can contemplate that the effect of the sizing technique is very similar either with a tetrahedron guided division or a dodecahedron guided division.

**[0058]**Reference is now made to FIG. 11 showing quite a large difference between elements of the present figure compared to elements of FIG. 9. This large difference is obtained simply by enlarging the associated geometrical figure contour radius r

_{d}. There are also three types of elements obtained by this division, pivoting elements 111, mobile elements 112, and gap elements 113. Puzzlewise, such a configuration is superior since a larger portion of the sphere's surface moves while playing the puzzle. So the fixed portion and the moving portion of a puzzle surface can be adjusted by the puzzle designer at will. This constitutes a big advantage when adapting the puzzle to different purposes, such as creating promotional vehicles, designing simple puzzle for kids or designing complex puzzles for the expert puzzle enthusiast.

**[0059]**Reference is now made to FIG. 12 in which a double-sliced circular dodecahedron face division applied to a sphere such as the one in FIG. 11 is shown. This double-slicing is carried with the same geometrical figure along the same guiding axis, except with two different radii r

_{d1}and r

_{d2}. It can be appreciated that a very interesting, symmetrical puzzle is obtained. This puzzle being constituted of six different types of elements, pivoting elements 121, mobile elements 122, gap elements 123, secondary mobile elements 124, secondary gap elements 125, and inter-gap elements 126. Here also the dedocahedron multi-slicing division is very similar to the tetrahedron multi-slicing division.

**[0060]**Reference is now made to FIG. 13 showing a circular icosahedron face division of a sphere at radius r

_{i}. Since the guiding polyhedron is an icosahedron there will be twenty pivoting elements 131, with the remaining of the puzzle's outer surface covered by mobile elements 132 and gap elements 133. Due to the great number of pivoting elements involved one can anticipate that the icosahedron family puzzles would be very challenging.

**[0061]**Reference is now made to FIG. 14 illustrating an application of the multi-pivoting technique applied to the icosahedron-based puzzle of FIG. 13, sliced at radius r

_{i2}. The number of different elements is now five, namely pivoting elements 141, mobile elements 142, gap elements 143, secondary mobile elements 144, and secondary pivoting elements 145. Great similarities exist with the previous puzzles. It is possible to anticipate a large magnitude of permutations resulting from the application of a mutli-slicing technique to such an icosahedron-based puzzle. Also, by introducing a multi-guiding technique to this family of puzzles, a countless number of puzzles could be obtained, and these would be almost impossible to solve unless appropriate visual indicia patterns were used to modulate (simplify) the difficulty level of these puzzles.

**[0062]**Reference is now made to FIG. 15 showing the proposed dividing method applied to a cubic solid. In this schematic representation of the dividing method, the given solid to be divided is represented by a cube C with an inscribed octahedron 0. For each face of the guiding polyhedron there is an associated axis f-f passing through a cube vertex, since the guiding octahedron is positioned such that its vertices are coincident with the geometrical centers of each cube face (point po). Not shown are associated geometrical figures F. The axis system is completed by having vertex axis v-v passing through the geometrical centers of each cube face, and midpoint edge axis e-e passing through cube edge midpoints. Also not shown are the associated geometrical figures V and E. The contours of the associated geometrical figures F, V, E are then used through projection along their respective axes to divide, or slice, the cubic solid into perfectly interfitting parts.

**[0063]**Reference is now made to FIG. 16 in which the associated geometrical figures F are six-pointed stars, the figures V are octagonals, and figures E are hexagonals used for dividing the cube-shaped solid. The preceding division is described as a single-sliced six-pointed star octahedron face to cube vertex division, with a single-sliced octagonal octahedron vertex to cube face division, with a single-sliced hexagonal octahedron edge to cube edge division of a cube. The resulting divided cube is not realistically intended to be implemented as a puzzle, but this however illustrates the extreme possibilities of the dividing method.

**[0064]**Reference is now made to FIG. 17 illustrating a single-sliced circular octahedron face to cube vertex division of a cube suited for puzzle purposes. This division being carried at radius r

_{c}. The resulting cube is of a completely novel aspect. The complete cube can be assembled from only four different types of elements, pivoting elements 171, mobile elements 172, secondary gap elements 173, and gap elements 174. Again many similarities can be observed with the preceding puzzles.

**[0065]**Reference is now made to FIG. 18 where a combination of a single-sliced circular octahedron face to cube vertex division with a mutli-pivoting single-sliced circular octahedron vertex to cube face division is shown and applied to a cube. These two divisions are carried at radius r

_{c1}and r

_{c}2. A very interesting cubic puzzle is than easily obtained by the application of the present dividing method. The resulting puzzle is constituted of six different types of elements, namely pivoting elements 181, mobile elements 182, secondary gap elements 183, gap elements 184, secondary pivoting elements 185, and secondary mobile elements 186. This puzzle can be easily complicated by applying other techniques available in the dividing method giving almost an infinity of possibilities.

**[0066]**Reference is now made to FIG. 19 illustrating an exploded view of an odd-shaped solid divided by a single-sliced tetrahedron face division. In this figure, only the pivoting elements are identified. It is to be noted that each one has a different form. There are four pivoting elements, since a single-sliced tetrahedron face division is used, element 191, element 192, element 193, and element 194. They are all obtained from a division with circular geometrical figure of radius r

_{h}.

**[0067]**Reference is now made to FIG. 20 showing an assembled view of the odd-shaped solid of FIG. 19. These two figures present the extended possibilities of the dividing method applicable to any odd-shaped solid.

**[0068]**It is to be understood that the same techniques for arranging the display of colours, emblems, logos or other visual indicia on the outer surfaces of the puzzles to modulate the difficulty level of the puzzles presented in the prior art are also applicable to any of the puzzles obtained through the application of the dividing method disclosed herein. Complex descriptions of evoluted patterns are not included in the present disclosure for the sake of simplicity, but are well within the scope of the technology introduced here and can be easily derived from the principles already disclosed in the prior art and applied to the puzzles resulting from the present dividing method. Different visual indicia patterns (e.g. colours, logos, emblems, symbols, etc.) can be used to modulate the difficulty level of the puzzles. In other words, different versions of a given puzzle can be provided for novice, intermediate or expert players, or even for kids.

**[0069]**It should be noted that advertising, corporate logos or team logos could also be placed onto the surfaces of the puzzles obtained by the application of the present dividing method to create promotional vehicles or souvenirs.

**[0070]**Also worth mentioning is that it is possible to add sliding movements to the pre-existing shifting movement to further complicate the puzzles. Slidable elements can be added to underlying shiftable elements as described Applicant's U.S. patent application Ser. No. 11/738,673. Generally, this is done by superimposing permutable sliding elements at the outer face of a given puzzle that slide in grooves in the underlying faces of said given puzzle to provide both shifting and sliding movements. Each superimposed sliding element slides in a curved track (the adjoining grooves) over the outer faces of non-sliding given puzzle elements along a circular slideway groove formed by adjacent grooves. Thus, adding sliding elements to a given shifting puzzle greatly increases the complexity of said given puzzle. Such given puzzle is now said to combine both shifting and sliding features.

**[0071]**All the aforesaid sliding modifications are analogous to the modifications introduced in Applicant's U.S. patent application Ser. No. 11/738,673, and therefore need not be repeated herein.

**[0072]**Other polyhedra of any kind could also be used as the guiding polyhedron for bisecting any given solid with the present dividing method, all without departing from the scope of the present invention. Likewise, the dividing method could also be applied to any polyhedron to achieve and create other interesting and challenging puzzles. Accordingly, the drawings and description are to be regarded as being illustrative, not as restrictive.

**[0073]**It will be noted that exact dimensions are not provided in the present description since these puzzles can be constructed in a variety of sizes.

**[0074]**While the puzzle elements and parts are preferably manufactured from plastic, these puzzles can also be made of wood, metal, or a combination of the aforementioned materials. These elements and parts may be solid or hollow. The motion of the puzzle mechanism can be enhanced by employing springs, bearings, semi-spherical surface knobs, grooves, indentations and recesses, as is well known in the art and are already well described in the prior art of shifting and sliding puzzles. Likewise, "stabilizing" parts can also be inserted in the mechanism to bias the moving elements to the "rest positions", as is also well known in the art.

**[0075]**It is understood that the above description of the preferred embodiments is not intended to limit the scope of the present invention, which is defined solely by the appended claims.

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