Abstract
A jigsaw puzzle and method of manufacture wherein incongruent pieces can be assembled into distinct interlocking arrangements to produce at least two different, often unrelated, pictures from markings on the upper surface of each piece.
Claims
1. A puzzle comprising: a plurality of pieces each having color, pattern, or color and pattern markings upon at least a first side, wherein at least two different said first side color markings, pattern markings, or color and pattern markings are present in the completed puzzle, each piece further comprising at least one engaging portion for allowing the piece to temporarily interlock with a corresponding engaging portion on an adjacent piece to form a connector pair, the pieces being shaped such that they interlock in a first arrangement defining a first surface upon which the different markings on the first side form a first mosaic image, and interlock in a second arrangement distinct from the first, defining a second surface upon which the different markings on the first side form a second mosaic image, wherein a majority of the pieces are incongruent and the puzzle comprises a plurality of connector pairs of different shapes.
2. The puzzle of claim 1 in which at least one piece has at least three engaging portions, said engaging portions meeting three different adjacent puzzle pieces on at least three points, said points being positioned with rotational symmetry about a center point equidistant from said points.
3. The puzzle of claim 1 in which at least one piece has four engaging portions, said engaging portions meeting four different adjacent puzzle pieces on four points, said points having fourfold rotational symmetry about a center point between said meeting points.
4. The puzzle of claim 1 in which a first piece has a marking upon a first side that is essentially the same in appearance as a marking upon a first side of a second piece.
5. The puzzle of claim 1 in which a first piece has a first side marked with a solid color, a second piece has a first side marked with said solid color and a third piece has a first side marked with a different solid color.
6. A method of manufacturing a puzzle comprising the steps of: (a) defining a plurality of pieces, each having at least a first side with markings thereon, such that at least: (i) one piece has color, pattern, or color and pattern markings on at least a first side such that the appearance of the markings is essentially the same upon rotation of the piece; or (ii) at least two pieces have color, pattern, or color and pattern markings on at least a first side of each such that the appearance of the markings is essentially the same upon exchange of the two pieces, (b) assigning each piece to a primary position in a primary arrangement; (c) assigning each piece to a secondary position in a secondary arrangement; (d) assigning engaging portions to each piece such that each piece fits into its primary position in the primary arrangement and fits into its secondary position in the secondary arrangement; (e) counting the number of ways in which interlocking pairs of pieces can fit together; (f) reducing the number of ways in which interlocking pairs of pieces can fit together by reassigning pieces to different secondary positions in the secondary arrangement and repeating steps (d) and (e); (g) repeating step (f) until a satisfactory minimum number of piece fits is achieved.
7. A method as claimed in claim 6 wherein step (f) comprises the sub-steps of: (a) randomly selecting a first piece; (b) randomly selecting a second piece: (c) if the first piece and second piece are the same piece, which can occur because the selecting process assumes replacement of the first piece before the randomly selecting a second piece takes place, assigning that piece a random rotation that maintains the general appearance of markings on its first side; (d) if the first piece and second piece are different but have substantially equivalent markings on their first sides, exchanging the location of the two pieces and assigning each piece a random rotation that maintains the general appearance of markings on the first side of each; (e) repeating steps (a) through (d) with the same piece pair with different rotations or by using different pairs; (f) continuing step (e) with a general tendency of reducing the probability of the number of ways in which interlocking pairs of pieces can fit together.
Description
BRIEF DESCRIPTION OF THE DRAWINGS
[0028] FIG. 1A shows a mosaic formed from colored pieces.
[0029] FIG. 1B shows another mosaic formed from the same colored pieces as FIG. 1A.
[0030] FIG. 2A shows the mosaic of FIG. 1A with each piece labeled by color and piece number and each edge of each piece assigned a style number.
[0031] FIG. 2B shows the mosaic of FIG. 1B formed from the same labeled pieces as FIG. 2A in an initial arrangement.
[0032] FIG. 3 shows the calculation of reduced style numbers according to the arrangement of pieces in FIG. 2B.
[0033] FIGS. 4A and 4B show the mosaics of FIGS. 2A and 2B, respectively, after substitution with reduced style numbers from FIG. 3.
[0034] FIG. 5 shows a reduction in number of fits by simulated annealing of the arrangement of pieces in FIG. 2B.
[0035] FIG. 6 shows an arrangement of pieces obtained by simulated annealing of the arrangement of pieces in FIG. 2B.
[0036] FIG. 7 shows the calculation of reduced style numbers according to the arrangement of pieces in FIG. 6.
[0037] FIGS. 8A and 8B show the mosaics of FIGS. 2A and 6, respectively, after substitution with reduced style numbers from FIG. 7.
[0038] FIGS. 9A and 9B show the mosaics of FIGS. 8A and 8B, respectively, with each edge shaped according to its style number and each piece colored according to the mosaics of FIGS. 1A and 1B, respectively.
[0039] FIG. 10A shows a mosaic formed from colored pieces to depict a baby panda.
[0040] FIG. 10B shows another mosaic formed from the same colored pieces as FIG. 10A to depict a mother panda.
[0041] FIG. 11 shows the mosaic of FIG. 10A with each piece labeled by color and piece number.
[0042] FIG. 12 shows the mosaic of FIG. 10B formed from the same labeled pieces as FIG. 11 in an initial arrangement.
[0043] FIG. 13 shows a reduction in number of fits by simulated annealing of the arrangement of pieces in FIG. 12.
[0044] FIG. 14 shows an arrangement of pieces obtained by simulated annealing of the arrangement of pieces in FIG. 12.
[0045] FIGS. 15A and 15B show the mosaics of FIGS. 11 and 14, respectively, with each edge shaped according to its style number and each piece colored according to the mosaics of FIGS. 10A and 10B, respectively.
DETAILED DESCRIPTION OF THE INVENTION
[0046] Construction of a two-way jigsaw puzzle begins with the selection of two related mosaics with the property that each mosaic can be formed by rearranging the pieces of the other. Thus each mosaic is a pictorial version of an anagrama word formed by rearranging the letters of anotherand might therefore be called an anagraph. The creation and selection of these mosaics is a matter of art, experiment, and judgment. Once two suitable mosaics are selected, applying the method of the present invention produces a set of jigsaw puzzle pieces that can form those two mosaics while preventing the formation of many unintended mosaics. The method operates by modifying each edge of each piece so that it fits with its intended mates and most pieces are made incongruent so that individual pieces are not generally interchangeable.
[0047] FIG. 1A shows a mosaic formed from square pieces with different colors indicated by different patterns. FIG. 1B shows another mosaic formed by rearrangement of those same pieces.
[0048] FIG. 2A shows the mosaic of FIG. 1A with an alphabetical label for each color and a numerical label for each piece that bears that color. Conventionally, the pieces are labeled from left to right and top to bottom, although the order of labeling is not essential to the method. Thus, A1 is the first piece of the first color encountered as the labeling commences, B1 is the first piece of the second color, B2 is the second piece of the second color, Cl is the first piece of the third color, and so on.
[0049] Furthermore, each edge of each piece in FIG. 2A is assigned a style number. Beginning with the first piece A1, sequential style numbers are assigned beginning with the top edge and proceeding clockwise: +1, +2, +3, +4. On the second piece B1 the style number sequence continues: +5, +6, +7. For the fourth edge of piece B1, rather than continuing the style number sequence with +8, it is assigned the style number opposite of style number +2 on the edge of the first piece A1 with which it mates: ?2. Then the sequence is resumed for the edges of the third piece B2: +8, +9, +10; and the fourth edge of piece B2 is assigned the style number opposite of style number +6 on the edge of the second piece B1 with which it mates: ?6. Assignment continues in the same manner for the remainder of the pieces, with each newly encountered edge assigned a sequential style number and each mated edge assigned the opposite style number of its mate.
[0050] Once the pieces are completely labeled in FIG. 2A they are placed into an arrangement corresponding to the mosaic of FIG. 1B. Since there are multiple pieces of each color, there is latitude in where to place each piece. Likewise, since the fundamental shape of each piece possesses rotational symmetry, there is also latitude in the rotational orientation of each piece. FIG. 2B shows an initial arrangement of the pieces from FIG. 2A formed according to a simple procedure: for each piece of the mosaic of FIG. 1B, the first available piece of the appropriate color from FIG. 2A is placed without rotation. Thus, piece Cl is placed in the top leftmost position since it is the first available piece of the appropriate color. Piece B1 is placed in the next position to the right. Since piece B1 has now been placed, the next position further to the right is filled with piece B2. The process continues until all the pieces have been placed to complete the arrangement. This order of placement is not essential to the method but merely fulfills a requirement that the pieces be placed into an initial arrangement corresponding to the mosaic of FIG. 1B.
[0051] The assignment of opposite style numbers to mated edges in FIG. 2A encoded the requirement that those edges be shaped complementarily so that they fit together when the mosaic of FIG. 1A is assembled. If a jigsaw puzzle were constructed so that each style number corresponded to a unique edge shape and opposite style numbers corresponded to complementary edge shapes, then the pieces of FIG. 2A would fit satisfactorily into the intended mosaic of FIG. 1A and no others. Therefore, the presence of matings between edges with non-complementary style number pairs in FIG. 2B indicates the need to reassign style numbers so that the pieces can fit satisfactorily into the mosaic of FIG. 1B as well as the mosaic of FIG. 1A. Some style numbers which are shown as unique in FIG. 2B must be made equivalent so that all the edges are complementarily mated in both FIGS. 2A and 2B.
[0052] FIG. 3 shows the calculation of equivalent style numbers according to the arrangement of pieces in FIG. 2B. The first column lists the style number of each edge that is mated in FIG. 2B with an edge having a style number other than its own opposite. For example, style number +3 is listed in the first column since it appears on an edge that is mated with an edge bearing style number ?22 rather than style number ?3. The second column lists the style number of the mated edge for each style number listed in the first column. The third column lists the opposite of each style number in the second column, that style number being necessarily equivalent to the style number listed in the first column since both are mated to the style number listed in the second column in either FIG. 2A or FIG. 2B. For example, the edge having style number ?22 is mated with the edge having style number +22 in FIG. 2A and with the edge having style number +3 in FIG. 2B. Therefore, style numbers +22 and +3 must be equivalent to each other so that they can both fit with that common mate.
[0053] The links of equivalent style numbers due to the sharing of mates between FIGS. 2A and 2B can lead to chains of equivalent style numbers as further style numbers are brought into association. For example, style numbers +3 and +22 are made equivalent by the mating of style numbers +3 and ?22 in FIG. 2B. Furthermore, style numbers +22 and +20 are made equivalent by the mating of styles +22 and ?20 in FIG. 2B. Therefore, style numbers +3, +22, and +20 must all be equivalent.
[0054] The fourth column of FIG. 3 lists all of the equivalent style numbers found by the following operation. First, list the style number from the first column. Second, list the equivalent style number from the third column. Third, check whether any row in FIG. 3 lists that equivalent style number in the first column and, if one does, then list the equivalent style number unless that equivalent style number has already been listed. Continue listing equivalent style numbers by repeating the third step until it ceases adding new style numbers to the list.
[0055] For example, in the row of FIG. 3 having style number +3 in the first column, style number +3 is the first one listed in the fourth column. Style number +22 is listed next since it appears in the third column as the first equivalent of style number +3. Then the row having style number +22 in the first column is referenced and style number +20 from the third column of that row is added to the list. Those steps are repeated to reference the row having +20 in the first column and add style number +18 from the third column of that row to the list. Those steps are repeated again to reference the row having style number +18 in the first column and add style number +10 from the third column of that row to the list. Repeating those steps once more references the row having style number +10 in the first column, but since style number +3 from the third column of that row was already listed the listing is ceased. The entire chain of steps moving forward through equivalents yielded style numbers +3, +22, +20, +18, and +10 in the fourth column.
[0056] For a further example, in the row of FIG. 3 having style number +1 in the first column, style number +1 is the first one listed in the fourth column. Style number ?7 is listed next since it appears in the third column as the first equivalent of style number +1. No row in FIG. 3 lists style number ?7 in the first column, so the listing of equivalents in the fourth column is ceased.
[0057] The fifth column of FIG. 3 lists all of the equivalent style numbers found by the following operation. First, list the opposite of the style number from the first column. Second, check whether any row in FIG. 3 lists that style number in the first column and, if it does, then list the equivalent style number from the third column of that row unless that equivalent style number has already been listed. Continue listing equivalent style numbers by repeating the second step until it ceases adding new style numbers to the list.
[0058] Finally, for each row in FIG. 3, the fourth and fifth columns are inspected to find the style number with the smallest absolute value. If that style number occurs in the fourth column then it is listed in the sixth column of that row. If that style number occurs in the fifth column then its opposite is listed in the sixth column. By way of the steps described above, the sixth column of each row in FIG. 3 now lists the style number with the smallest absolute value that is equivalent to the style number listed in the first column of that row.
[0059] For example, in the row of FIG. 3 having style number +3 in the first column, the absolute values of the style numbers listed in the fourth column are 3, 22, 20, 18, 10; and the absolute values of the style numbers listed in the fifth column are 3, 10, 18, 20, 22. The smallest of these is 3. Since +3 occurs in the fourth column, it is listed in the sixth column as the reduced value for style number +3. Likewise, in the row having style number +14 in the first column, 9 is the smallest absolute value among the style numbers listed in the fourth and fifth columns. Since ?9 appears only in the fifth column, its opposite, +9, is listed in the sixth column as the reduced value for style number +14.
[0060] FIG. 4A shows the mosaic of FIG. 2A with each style number replaced by its reduced style number from the sixth column of FIG. 3. For any style number not appearing in the first column of FIG. 3 but whose opposite does appear, that style number is instead substituted with the opposite of the reduced style number of its opposite. Style numbers that are listed neither as themselves nor as their opposites in the first column of FIG. 3 are left unchanged. FIG. 4B shows these same operations applied to the mosaic of FIG. 2B.
[0061] By having replaced initial style numbers with reduced style numbers, FIGS. 4A and 4B show a consistent set of labeled pieces such that the style number of each edge is the opposite of any mated edges in both mosaics. The process of reducing style numbers to obtain this consistency also produced some duplication of style numbers. For example, in FIG. 4A there are five edges with style number +3 and five edges with style number ?3.
[0062] Since opposite style numbers indicate edges that are allowed to fit together, the number of fits for any edge of a piece is the number of edges on other pieces having the opposite style number of that edge. Other edges on the same piece having the opposite style number are not counted as fits since a piece cannot be positioned to mate two of its own edges with each other. If the pieces of FIG. 4A were separated and shuffled, then a user of the puzzle would find that the bottom edge of piece A1 having style number +3 could fit with edges having style number ?3 appearing on five other pieces: A2, B4, A4, C3, and C4. But only piece A2 is intended to mate with that edge of piece A1 in forming the mosaic of FIG. 1A, so the excess of fits makes finding the correct mate more uncertain and diminishes the user's feeling of accomplishment in mating two puzzle pieces together. Fitting piece A1 with either of pieces B4, A4, or C3 would be misleading in an attempt to construct the intended mosaic.
[0063] The population of misleading fits can be characterized by calculating the total number of fits for a set of puzzle pieces. The total number of fits for the pieces of FIG. 4A is calculated by first summing, for each edge of each piece, the number of appearances of the opposite of that edge's style number on the edges of any other pieces. That sum is then halved to avoid double counting each mating pair, yielding a total count of 47 fits. An identical count can be calculated for the pieces of FIG. 4B since they are merely the same pieces as FIG. 4A in a different arrangement.
[0064] It may be desirable to reduce the number of fits for a set of puzzle pieces so that the puzzle user experiences less frustration in assembling the intended mosaics and gains the satisfaction that once a fit is found it is likely to be a sign of progression toward completion of whichever mosaic the puzzle user is attempting. To achieve that goal, the number of fits for a set of puzzle pieces is minimized by the well-known optimization method of simulated annealing.
[0065] Simulated annealing mimics the physical process of thermal annealing in which a material is heated and then slowly cooled to reach an ideal state; e.g., to achieve the lowest energy state of the system at a given temperature. As a computational method it optimizes a system by defining an energy function to measure how close the system is to an ideal state and then randomly changing the state of the system with a bias toward accepting changes that reduce that energy. A fictitious temperature having units of energy adjusts the magnitude of the bias and by gradually reducing that temperature from high values to low values the system is allowed to explore a wide variety of candidate states and then settle gradually into one with a minimum of energy. The method is especially useful when the number of possible states of a system is too large to be explored exhaustively and an optimized state is sought with less effort. For the present invention, simulated annealing is enacted by defining the number of fits as the measure of energy, rearranging pieces in the second mosaic to change the state, and applying a fictitious temperature to find an arrangement minimizing the number of fits.
[0066] It should be noted that the arrangement of pieces in FIG. 2B involved considerable latitude in the selection of a piece and rotation to fill each location with a piece of the appropriate color. To begin altering the arrangement, a first piece is chosen at random and a second piece is chosen at random. If the two pieces are different colors then they are each left unchanged. But if the two pieces chosen are the same color, then their positions are swapped and a random rotation of 0, 90, 180, or 270 degrees is given to each piece. By chance the same piece might be chosen as both the first and second piece, in which case that piece is left in place but given a random rotation. After making any changes, the number of fits is recalculated by the same methods employed for the initial arrangement and compared to the number of fits for the prior arrangement.
[0067] If the number of fits decreased or stayed constant then the new arrangement is kept as the basis for another change. If the number of fits increased then the new arrangement is kept with a probability equal to the mathematical constant e (approximately 2.71828) raised to the power of minus the ratio of the increase in number of fits to the temperature for that calculation. Otherwise the change is undone. This algorithm tends to accept increases in the number of fits that are small relative to the temperature and reject increases that are large relative to the temperature. But even large increases in the number of fits are sometimes accepted so that significant changes in the overall arrangement can be considered.
[0068] After the change is attempted and either kept or rejected, the resulting arrangement is used as the reference state for another change. The process is repeated many times while decreasing the temperature from a value that produces high probabilities of acceptance to one that produces low probabilities of acceptance. A computer may be employed to automate the process of generating changes, calculating the number of fits, and deciding whether to keep or undo each change. (See, e.g., Simulated Annealing Methods in Numerical Recipes, William Press; Saul Teukalsky, William Vettering, Brian Flannery, ed., Third Edition 2007.) FIG. 5 shows the variation in the number of fits while the temperature was decreased from 10000 to 0.1 over the course of 600 million attempted changes. Note that at high temperatures the number of fits sometimes rose above the initial value but upon cooling settled to much lower values. FIG. 6 shows a rearrangement of pieces from FIG. 2B obtained by simulated annealing. Random rotations have left some of the pieces in different orientations than in FIG. 2B as indicated by the rotated labels, but the style numbers assigned to each edge of each piece are identical to those assigned to the corresponding pieces in FIG. 2B.
[0069] FIG. 7 shows the calculation of reduced style numbers according to the arrangement of FIG. 6 in the same manner as FIG. 3 was calculated from the arrangement of FIG. 2B. FIG. 8A shows the mosaic of FIG. 2A with each style number replaced by its reduced style number from FIG. 7 or by the opposite of the reduced style number of its opposite. FIG. 8B shows the same operations applied to the mosaic of FIG. 6. The puzzle pieces shown in FIGS. 8A and 8B allow only 27 fits, a substantial improvement from the 47 fits for the pieces in FIGS. 4A and 4B derived from the initial arrangement of FIG. 2B. Since each of the mosaics in FIGS. 8A and 8B includes 17 fits which are necessary to form the intended puzzle shapes, the pieces allow only 10 fits in excess of the minimum which must exist for either mosaic alone. Consequently, the puzzle user will experience a marvelous sensation that the puzzle is guiding him toward whichever mosaic he is attempting to assemble without offering many misleading and frustrating false fits.
[0070] Having reduced the number of fits for the set of puzzle pieces used to produce the mosaics of FIGS. 8A and 8B, the style numbers are used to guide the shaping of each puzzle piece. A curve defining two complementary edges is chosen for each pair of positive and negative style numbers. Each curve spans between two points corresponding to two successive corners of a piece. The curves may be shaped so that the edges interlock to resist being pulled apart by motion in the plane of the puzzle pieces. Different curves are chosen for different style number pairs so that the edges defined by each curve do not fit satisfactorily with the edges defined by any other curve. Unsatisfactory fits may include pairs of edges that overlap or leave uneven gaps when placed together.
[0071] FIGS. 9A and 9B show the mosaics of FIGS. 8A and 8B, respectively, with each edge shaped according to the curve defined for its style number. In this embodiment, the curves defining the edges were chosen so that positive style numbers formed protruding tabs and negative style numbers formed recessed blanks. However, it may be desirable to change the polarity of some curves to hide evidence of the design method from the puzzle user or to produce a more pleasing balance of tabs and blanks on most of the pieces. A puzzle user attempting to assemble the pieces of FIGS. 9A and 9B will find that the pieces can form into the arrangements of FIGS. 9A and 9B but resist assembly into other cohesive arrangements.
[0072] In a second embodiment of the invention, FIG. 10A shows a first mosaic depicting a baby panda. FIG. 10B shows a second mosaic formed by rearrangement of the pieces from the first mosaic to depict a mother panda. The process to design puzzle pieces for forming the mosaics of FIGS. 10A and 10B is identical to the process used to design puzzle pieces for forming the mosaics of FIGS. 1A and 1B. The steps of that process are merely extended to a larger number of pieces.
[0073] The pieces of the mosaic of FIG. 10A are first labeled by color and number as shown in FIG. 11. Then style numbers are as assigned to each edge in the same manner as the first embodiment. Those labeled pieces are placed into an initial arrangement as shown in FIG. 12 corresponding to the mosaic of FIG. 10B. Reduced style numbers are calculated and substituted for the initial style numbers. Then the number of fits is calculated, yielding a count of 1762 fits for the initial arrangement. FIG. 13 shows the variation in the number of fits while the temperature was decreased from 10000 to 0.1 over the course of 5.4 billion attempted changes. At the end of simulated annealing the pieces were arranged in the manner of FIG. 14, reducing the number of fits to 454. FIGS. 15A and 15B show the mosaics of FIGS. 11 and 14, respectively, with each edge shaped according to its style number after annealing and substitution.
[0074] The embodiments described are for illustration only and it will be readily understood that the invention is not limited to these embodiments. Some variations keeping in spirit with the invention are disclosed below.
[0075] The design of each puzzle begins with a pair of mosaics having the same number of pieces, the same number of colors, and the same number of pieces of each color. The two mosaics may have the same overall shape and dimensions, but they may also differ in shape and dimension so long as the numbers of pieces are consistent. The overall shapes need not be rectangular and may form an irregular arrangement of connected pieces. The pieces of each mosaic need not all be connected and may form two or more separate subsections. The number of different colors used for each puzzle is limited only by the number of pieces in that puzzle. Moreover, the invention extends to puzzles with any number of pieces.
[0076] The embodiments described have used a basis of substantially square elements within the mosaics to define the pieces. But any form of polygon is acceptable including, but not limited to, rectangles, triangles, rhombuses, and hexagons. During thermal annealing, random rotations are applied according to whatever degrees of rotational symmetry the basis polygons possess. Moreover, a puzzle need not comprise only a single class of basis polygon. If more than one class of basis polygon is employed then during annealing changes are allowed only when the two randomly selected pieces belong to the same class. For whatever basis polygons are employed, the shaping stage consists of applying a different curve to each style number pair.
