«Contents Ü Foreword Elwyn Berlekamp and Tom Rodgers ½ I Personal Magic ¿ Martin Gardner: A “Documentary” Dana Richards ½¿ Ambrose, Gardner, ...»
Colormaze: Square tiles in six colors form double polyomino shapes with no duplicate color in any row, column, or diagonal. In another use, double polyominoes can be formed with ¾¢¾ quadrants of each color and then dispersed through a maze-like sequence of moves to the desired ﬁnal position (Figure 6).
THOSE PERIPATETIC PENTOMINOES 111
Sextillions: the hexomino shapes can form double through sextuple copies of themselves and various enlargements of the smaller polyominoes. Sizecompatible with Poly-5.
Snowﬂake Super Square: the 24 tiles are all the permutations of three contours—straight, convex, concave—on the four sides of a square. They can form various single- and double-size polyomino shapes (Figure 8).
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Triangoes: orders 1 and 2 polytans permuted with two or more colors (square, triangle, parallelogram). The pieces can form diagonally doubled pentominoes and hexominoes with colormatching adjacency of tiles (Figure 9).
Lemma: A matrix of multiple grids lends itself to polyomino packing with checkers in three colors.
Multimatch I: The classic MacMahon Three-Color Squares have been discovered to form color patches of pentominoes and larger and smaller polyominoes within the ¢ rectangle and ¢ square, including partitions into multiple shapes (Figure 10).
THOSE PERIPATETIC PENTOMINOES 113
Multimatch II: The 24 tri-color squares with vertex coloring (each tile is a ¾ ¢ ¾ of smaller squares) can form polyominoes both as shapes with colormatching and as color patches on top of the tiles (Figure 11).
Gallop: On a ¢ ½¾ grid, double checkerboard hexomino shapes deﬁned by pawns are transformed with knight’s moves (borrowed from the Leap set). Another puzzle is to change a simple hexomino made of six pawns into as many different other hexominoes as possible by moving the pawns with chinese-checker jumps (Figure 12).
Tiny Tans: Four triangle-based pieces can make some pentomino shapes.
The T and U puzzles are actually dissected pentominoes (Figure 13).
Throw a Fit: Mulit-colored cubes form pairs of pentominoes with colormatching.
Perplexing Pyramid: A Len Gordon invention, the six pieces comprising 20 balls can make most of the pentominoes in double size.
Quantum: Pentomino and larger polyomino shapes are created by movement of pawns (the puzzles to be published as a supplement to the existing game rules).
Rhombiominoes: A skewed embodiment of pentominoes, where each square is a rhombus the size of two joined equilateral triangles. The 20 distinct pieces form a ½¼ ¢ ½¼ rhombus (Figure 14). This is a limited-edition set.
Figure 14.THOSE PERIPATETIC PENTOMINOES 115
Heptominoes: These sets of the 108 seven-celled polyominoes are produced by popular demand, sized to Sextillions and Poly-5.
Octominoes: The 369 eight-cell shapes exist in limited-edition sets, sized to smaller members of the family.
The Hexacube: The 166 planar and non-planar hexacube pieces plus 4 unit cubes form a ½¼¢½¼¢½¼ cube. A vast territory just begging for exploration, this set is sized to Quintillions.
The above concepts arose and became possible because Martin Gardner fostered a love of mathematical recreations among his readers. A natural side-effect of his raising the consciousness of the public about the joys of combinatorial sets has been the proliferation of other sets, besides polyominoes: polyhexes, polyiamonds, polytans. All these are ﬁnding homes in the Kadon menagerie.
Throughout the years of Kadon’s evolution, the inspiring and nurturing presence of Martin Gardner has been there, in the reprints of his columns, through articles in various publications, with kind and encouraging words in correspondence, and always helpful information. We were honored that Martin selected us to produce his Game of Solomon and the Lewis Carroll Chess Wordgame, using Martin’s game rules. The charm, humor and special themes of these two games set them apart from all others, and we are dedicated to their care and continuance.
In retrospect, then, my personal career and unusual niche in life came about directly as a result of the intellectual currents and eddies created by one mind—Martin Gardner’s—whose ﬂow I was only too happy to follow.
I cannot imagine any other career in which I would have found greater satisfaction, fulﬁllment and never-ending challenge than in the creation of
beautiful playthings for the mind. And assuredly, if there is a 90-degree angle or parallelogram to be found, the pentominoes will announce themselves in yet another manifestation. Thank you, Martin Gardner.
Self-Designing Tetraﬂexagons Robert E. Neale Flexagons are paper structures that can be manipulated to bring different surfaces into view. The four-sided ones discussed here are “ﬂexed” by folding them in half along an axis, and then opening them up a different way to reveal a new face. Sometimes the opening is in a different direction (mountain or valley fold), sometimes along the other axis, and sometimes both.
Although usually constructed from a strip of paper with attachment of the ends, ﬂexagons seem more elegant when no glue or paste is required. Interesting designs of the faces can be found when the material is paper, or cardboard, with a different color on each side (as in standard origami paper). I call these “self-designing.” What follows is the result of my explorations.
Although the text refers only to a square, the base for construction can be any rectangle. There are ﬁve different starting bases (paper shapes; see Figures 1–5). There are three different ways of folding the bases. Warning: Not all the results are interesting. But some are, my favorites being three: the ﬁrst version of the cross plus-slit (for its puzzle difﬁculty); the ﬁrst version of the square slash-slit (for the minimalist base and puzzle subtlety); and the third version of the square cross-slit (for its self-design). Note that the text assumes you are using two-colored paper, referred to, for simplicity, as black and white.
Square Window The ﬁrst ﬂexagon I saw that did not need glue was shown to me by Giuseppi Baggi years ago. It is a hexa-tetraﬂexagon made from a “window” — a square (of 16 squares) with the center (of 4 squares) removed (Figure 1). It has six faces that are easy to ﬁnd. ½ I have recently used this window model for a routine about the riddle of the chicken and egg: “What’s Missing Is What Comes First.” This involves decorating the paper with markers, so is a separate manuscript.
It should be noted, however, that there are two ways of constructing this ﬂexagon, both of which are discussed below.
Same Way Fold. As the dotted line indicates in Figure 6, valley fold the left edge to the center. The result is shown in Figure 7. As the dotted line indicates in Figure 7, valley fold the upper edge to the center. The result is ½Directions for constructing the hexa-tetraﬂexagon are found on pages 18–19 of Paul Jackson’s Flexagons, B.O.S. Booklet No. 11, England, 1978.
SELF-DESIGNING TETRAFLEXAGONS 119 shown in Figure 8. As the dotted line indicates in Figure 8, valley fold the right edge to the center. The result is shown in Figure 9.
The ﬁnal fold is very tricky to communicate, but not at all tricky to do when you understand it. The goal is to make this last corner exactly the same as the other three. As you would expect, the bottom edge will be valley folded to the center. The right half of this lower portion falls directly on top of the portion above it. The left half of this lower portion goes underneath the portion above it. (So Figure 9 shows a valley fold on the right half and a mountain fold on the left half.) To make this actually happen, lift up the upper layer only of the left half of the lower portion, and then fold the entire lower portion to the center. The result is shown in Figure 10.
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Note that the model is entirely symmetrical. The four corners are identical, and both sides of the model are identical. (If this is not the case, you have made a mistake, probably on the last move.) This method amounts to folding the edges to the center, one after another (proceeding either clockwise or counterclockwise around the square), all of the folds being valley folds. This produces a Continuous Flex, moving in a straightforward manner from face 1 to 2 to 3 to 4 to 1, 5, 6, 1. The designs procured are merely three black faces and three white faces, as follows: black, black, white, white, black, black, white, white, black.
Alternating Way Fold. As Figure 11 indicates, mountain fold the left edge to the center. The result is shown in Figure 12. As the dotted line indicates in Figure 12, valley fold the upper edge to the center. The result is shown in Figure 13. As the dotted line indicates in Figure 13, mountain fold the right edge to the center. The result is shown in Figure 14.
Before you make the ﬁnal valley fold of the bottom edge to the center, pull the bottom layer at the lower left corner away from the upper layer.
Make the valley fold, and allow that bottom layer to go back to the bottom of the lower left corner. The result is shown in Figure 15. Note that opposite corners are identical, and both sides of the model are identical.
This noncontinuous Puzzle Flex can be ﬂexed continuously through four faces only: 1 to 2 to 3 to 4 to 1, etc. Finding the other two faces is easy in this
particular case, but backtracking is required. The four faces are identical:
a checkerboard pattern of two black squares and two white squares. The other two faces are a solid black face and a solid white face.
Cross Plus-Slit The window model inspired me to think of other ways to form a ﬂexagon without having to attach the ends to each other. One result was a puzzle made from a “cross” — a square (of 16 squares) with the four corner squares removed, and two slits in the center that intersect in the shape of a plus sign (see Figure 2).
This base is manipulated in a third way, the 3-D Tricky Way. This puzzle is interesting for two reasons: It is tricky to construct, and, while four of the faces are easy to ﬁnd, the other two faces are quite difﬁcult to discover, involving changing the ﬂat ﬂexagon into a three-dimensional ring and back again. The four faces are continuous and identical, being black and white checkerboard patterns. The other two faces are solid black and solid white.
(Note: The trick fold for constructing the ﬂexagon can be done in a slightly different way that has the four faces not continuous.) 3-D Tricky Fold In order to follow the instructions and understand the diagrams, number the base from one to six, exactly as indicated in Figure 16.
Orient them just as indicated. The numbers in parentheses are on the back side of the base. You will make two, very quick, moves. The ﬁrst renders the base 3-D, and the second ﬂattens it again.
Note the two arrows in Figure 16. They indicate that you are to make the base 3-D by pushing two opposite corners down and away from you. So reach underneath and hold the two free corners of the 1 cells, one corner in each hand. When you pull them down and away from each other, they are turned over so you see the 5 cells. The other cells come together to form two boxes open at the top. There is a 6 cell at the bottom of each box (see
Figure 17). Now the base should be ﬂattened into a compact model with four cells showing on a side. Change your grip so you are holding the two inside corners of the 5 cells. (These are the corners opposite the ones you were holding.) Now push down on these corners, and at the same time, pull them away from each other. The 3-D boxes will collapse, the base forming a ﬂat square of four cells. Once this happens, you should have four 1 cells facing you, and four 4 cells on the other side. The result is shown in Figure 18.
Sometimes, however, you will ﬁnd that another number has appeared, a 2 instead of a 1, or a 3 instead of a 4. You can correct this easily by tucking the wrong cell out of sight, replacing it with the proper one. Check both sides. The model is completed.¾ Flexing. You can ﬁnd the faces numbered 1 to 4 by the usual ﬂexing. Faces 5 and 6 are found by the following procedure. Begin with face 1 on the top and face 4 on the bottom. Mountain fold the model in half on the vertical axis, the left and right halves going back away from you. Do not open the model as you do when ﬂexing. Rather, move the lower inside packet of squares (with 2 and 3 on the outside) to the left, and the upper inside packet of squares (with 2 and 3 on the outside also) to the right. Now open the model into a tube — a cube open at both ends. Collapse the tube in the opposite way. This creates a new arrangement. Flex it in the usual way to show face 5, then 3, and then make the tube move again to return to face 4 ¾Other directions for constructing this ﬂexagon are on page 27 of Jackson’s Flexagons, and in Martin Gardner’s Wheels, Life and Other Mathematical Amusements, New York: W.H. Freeman & Co., 1983, pp. 64–68.
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and then face 1. To ﬁnd face 6, begin on face 4, with face 1 at the back, and repeat the moves just given.
Another version of this ﬂexagon can be made by using the Same Way Fold. (Alternating Way gets you nowhere.) Four faces are easily found, but they are not continuous. Two faces can be found only by forming the rings, as mentioned above. The designs of the four faces are two solid black and two checkerboard. The other two faces are solid white.