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Most intriguing about this puzzle, and the most difﬁcult aspect to investigate was the fourth observation: the color combinations need not be as shown in Figure 1. The previous question now becomes more interesting.

How many solutions are there for each color combination? And how many color combinations are there?

This paper addresses these last two questions. The individual dicubes can be colored thirty-six different ways. Twelve of these can be selected in 1,251,677,700 (36!/12!24!) different ways of which, as determined in Section 4, only 133,105 of these combinations meet the three-cubes-of-each-color requirement. Since one person can assign the fourth and ﬁfth colors to orange and red and another person to red and orange, it becomes apparent that most of the 133,105 combinations are isomorphisms. It becomes necessary, then, to separate these cases into disjoint sets of isomorphisms. Fortunately, as is shown in Section 5, only thirty dicubes are required, and only 10,691 combinations need to be sorted into disjoint sets.

The method of sorting the 10,691 combinations into 148 disjoint sets is described in Section 6, in which the number of isomorphisms using the set of thirty-six dicubes is also determined.

The ﬁnal part of this analysis was to determine the number of puzzle solutions for each disjoint set of dicube combinations. This was achieved through the use of a computer program; the results are shown in Table 1.

THE NINE COLOR PUZZLE 153

1. Additional Comments The Kolor Kraze puzzle shown in Reference 1 and the nine color puzzle shown in Reference 2 are isomorphic, thus leading me to believe that they originated from the same source.

It is not necessary that the tricube be straight, and, as mentioned in Section 2, an irregular tricube may have three locations. The number of solutions for these cases are included in Table 1.

There are two rather startling phenomena shown in Table 1. While some color combinations within the set of thirty dicubes have up to 720 isomorphisms, there are two color combinations (Numbers 73 and 121) which, no matter how the colors are swapped, map onto themselves. Verifying this would be an interesting exercise for the reader. Stranger yet, these two cases have no puzzle solutions. Is this a coincidence?

2. Permissible Locations of the Tricube as Determined by Parity Restrictions

The three layers of the cube are checkered as shown in Figure 2, where odd numbers represent one state and even numbers the other. Each dicube when similarly checkered has one even and one odd number. Since the cube has fourteen odd numbers and thirteen even numbers, the tricube must ﬁll two odd numbered spaces.

There are only two possibilities. The tricube location ﬁlling spaces 1, 10, and 19 is referred to as an “edge case” while the tricube location in spaces 5, 14, and 23 is referred to as a “center case.” Other possible locations are reﬂections or rotations of these and are not considered as separate puzzles solutions.

158 S. FARHI If an irregular tricube is used, there are three possible locations. The corner case ﬁlls spaces 1, 2, and 11; the edge case ﬁlls spaces 2, 5, and 11; the center case ﬁlls spaces 5, 11, and 14.

3. The “No Two in the Same Plane” Rule

One cube is centrally located and thus not visible. The only way the remaining two cubes of that color can be displayed on all six faces is for them to be located on opposite corners as shown in Figure 3 by the letter “A.” The remaining six corners must have cubes of different colors. Since the corner colors are displayed on three faces, the remaining two cubes must be located on an edge, accounting for two faces, and on a centerface such as shown by the letter “B.” The remaining two colors must all be on edges, such as shown by the letter “C.” In each case, cubes of the same color are never in the same plane.

4. Determination of the 133,105 Possible Dicube Color Combinations The tricube colors are deﬁned as colors 1, 2, and 3. The two dicubes with color 1 may use any two colors in the ﬁrst row of the following table. If color 2 is selected from the ﬁrst row, then a third dicube with color 2 and any color from the second row is selected. If color 2 is not selected from the ﬁrst row then color 2 is combined with any two colors from the second row for the third and fourth dicubes.

If none of the previous selections include color 3, then color 3 is combined with two colors from the third row. If color 3 has been selected once, then one color is selected from the third row and if color 3 has been selected twice, then none are selected from the third row.

THE NINE COLOR PUZZLE 159

** First Color Second Selection **

The process continues: color 4 is combined with colors in the fourth row such that there are three cubes with color 4, color 5 is combined with colors in the ﬁfth row such that there are three cubes with color 5, etc.

Using this algorithm, a computer program established 133,105 possible color combinations.

5. Reduction of the Number of Isomorphisms The table in Section 4 is simpliﬁed by restricting the selection in the ﬁrst two rows as shown below.

This can be justiﬁed by deﬁning the colors associated with color 1 that are not 2 or 3 as colors 4 and 5. Similarly if color 2 is not associated with colors 3, 4, or 5, the two new colors are identiﬁed as colors 6 and 7. This results in requiring only thirty cubes and, using the same algorithm as before, only 10,691 color combinations.

cubes of each color requirement and the restriction that the second number be greater than the ﬁrst.

Identiﬁcation was further simpliﬁed by placing these numbers in pairs (see Note below), i.e., (24), (35), (67), (68), (98), and (99), and then assigning a printable character to each, by adding twenty-three to each and using the ASCII character set codes (ASCII stands for American Standard Code for Information Interchange). Thus this color combination is identiﬁed as /:Z[yz.

The use of this six-character identiﬁcation greatly simpliﬁed the process of sorting the 10,691 cases into 148 disjoint sets of isomorphisms.

Decoding a name is quite simple. For example, the ﬁrst entry in Table 1 is.:Z[yz. The ASCII codes for these characters are: 46, 58, 90, 91, 121, and

122. Subtracting twenty-three results in: 23, 35, 67, 68, 98, and 99. It is then a simple process to recognize that this represents the dicubes colored (1, 2), (1, 3), (2, 3), (4, 5), (4, 6), (4, 7), (5, 6), (5, 8), (6, 9), (7, 8), (7, 9), and (8, 9).

Any permissible interchange of colors is an isomorphism. Color 2, which is in the center of the tricube, cannot be interchanged with other colors. Colors 1 and 3, on the ends of the tricube, can be swapped with each other, but not with any of the other colors. The remaining six colors may be interchanged in numerous ways: two at a time, three at a time, four at a time, including pairs of two at a time, ﬁve at a time including three at a time with two at a time, and six at a time including a triplet of two at a time, pairs of three at a time and four at a time with two at a time. Not all of these mappings produce a new isomorphism.

Note: The smallest possible number is 23 and the largest is 99. The ASCII code for 23 is not a printable character. An inspection of the ASCII table will explain why it was decided to add 23.

The 10,691 color combinations obtained by the algorithm described in Section 5 were listed in order according to their ASCII characters. The ﬁrst entry,.:Z[yz, has sixty isomorphisms. These were removed from the list.

The head of the list then became.:Zppp; its isomorphisms are determined and removed from the list. The process was then continued until the list was exhausted. This process then identiﬁed the 148 disjoint sets of isomorphisms.

Shown in Figure 4 are an edge solution and a center solution for the color combination of Figure 1. The solutions are identiﬁed by a three-digit number indicating the color of the buried cube and the two “edge” colors.

Different solutions often have the same three-digit identiﬁcation. If the solutions are to be catalogued, then additional criteria for recording solutions are recommended.

THE NINE COLOR PUZZLE 161

** Figure 4. Typical solutions.**

The edge solution is interesting in that new solutions are often obtainable by using two transformations. Whenever the tricube is on an edge and shares a plane with only three dicubes, the plane can be translated to the other side resulting in a new solution. Often two dicubes may be swapped with two others having the same colors. For example, dicubes (4, 7) and (6,

9) in the bottom layer may be swapped with dicubes (4, 6) and (7, 9) on the upper right. Using these transformations, the reader should now be able to determine ﬁve more edge solutions.

Stan Isaacs has suggested that graph analysis would be useful in determining the number of solutions. A preliminary investigation shows some merit in utilizing graph analysis to illustrate why some color combinations have no solutions while other color combinations have numerous solutions.

Unfortunately, all 148 graphs have not been compared.

Puzzle sets or puzzle solutions may be obtained by contacting the author by E-mail at sivy@ieee.org.

References [1] Slocum, Jerry, and Botermans, Jack. Creative Puzzles of the World. Harry N.

Abrams, New York, 1978. 200 pp., hardcover.

[2] Meeus, J., and Torbijn, P. J. Polycubes. Distracts 4, CEDIC, Paris, France, 1977. 176 pp., softcover, in French.

Twice: A Sliding Block Puzzle Edward Hordern Twice is a new concept in sliding block puzzles: Some blocks are restricted in their movements and can only reach certain parts of the board from particular directions or, in some cases, cannot get there at all.

The puzzle was invented by Dario Uri from Bologna, Italy, and was originally issued in 1989 with the name “Impossible!!” Subsequent additions and improvements made over the next couple of years led to a change of name, to “Twice.” The name was chosen because there are two quite different puzzles involving two different blocks numbered 2(a) and 2(b), one being used in each puzzle.

Description Fixed into the base of the board are four pegs, one in each corner (see Figure 1).

There are nine square blocks (including two numbered 2), but only eight are used in each puzzle. Channels (or grooves) are cut into the bases of some blocks, allowing them to pass over the pegs in the corners (see Figure 2). Blocks can either have a horizontal channel, a vertical channel, both channels, or no channel at all. Blocks A, 7, and 2(b) have a single horizontal channel, and these blocks can only reach the corners from a horizontal Written by Edward Hordern and reproduced with the kind permission of Dario Uri.

164 E. HORDERN direction. Blocks 4 and 2(a) have a single vertical channel and can only approach the corners from a vertical direction. Blocks 1 and 5 have both channels (in the form of a cross) and can go anywhere. Blocks 3 and 6 have no channels and cannot go into any corner.

The Puzzle Figure 3 shows the start position. The ﬁrst puzzle uses block 2(a), and the second puzzle uses block 2(b). The object of the puzzle in each case is to move block A to the bottom right corner. The puzzles are both rated as difcult, the second being the harder of the two. It is quite an achievement just to solve them. For the expert, however, the shortest known solutions are 50 moves for the ﬁrst puzzle and a staggering 70 moves for the second. Both solutions are believed (but not proved) to be minimum-move solutions.

The delight of the ﬁrst puzzle is that it is very easy to move block A to just above the bottom right corner, only to get hopelessly stuck…. So near, and yet so far! In puzzle 2 it is quite a task to move block A more than a square or two, or to get anywhere at all!

Hints for Solving In both puzzles the “nuisance” blocks are 3 and 6. Since they can’t go into the corners, they must only move in a cross-shaped area. During the solution they must continually be moved “around a corner” to get them out of the way of another block that has to be moved. Once this has been mastered, a plan can be made as to which blocks have to be moved so as to allow block A to pass. The real “problem” blocks are block 7 in the ﬁrst puzzle and blocks 7 and 2(b) in the second. These blocks, as well as block A, can only move freely up and down the center of the puzzle. This causes something of a trafﬁc jam, which has to be overcome….

Planar Burrs M. Oskar van Deventer Usually “burrs” are considered to be three-dimensional puzzles. The most common are the six-piece burrs, which occur in many different designs.

The most interesting are those with internal voids, because these can be so constructed that several moves are needed to separate the ﬁrst piece. Since about 1985 a “Most Moves Competition” for six-piece burrs has been running. Bill Cutler’s “Bafﬂing Burr” and Philippe Dubois’ “Seven Up” were the ﬁrst attempts. I am not completely aware of the state of the competition, but as far as I know Bruce Love is the record holder with his “Love’s Dozen,” which requires twelve moves!

Recently, I received from Tadao Muroi, in Japan, an ingenuously designed puzzle that he called “Four Sticks and a Box.” The puzzle has no more than four movable pieces, but nevertheless requires twelve moves (!) to get the ﬁrst piece out of the box. Muroi wrote me that his idea was inspired by a design of Yun Yananose, who was inspired by “Dead Lock,” a puzzle of mine. Though Muroi’s puzzle is three-dimensional, all interactions between the four pieces take place in one plane, so in some sense it might be considered a planar burr. However, it cannot be realized as a planar burr because in two dimensions each piece should be disconnected.