Mixed Angle

New Polyform Possibilities Using Mixed Angle Systems

Previously, puzzles have tended to confine themselves to only one angle system. For example, the 12 pentominoes (pieces built of five squares) use only the 90° (“ortho-“) system, while Stewart Coffin’s “Snowflake” puzzle (pieces built of three or four hexagons, called “trihexes” and “tetrahexes”) uses a uniform 60° system. This is perfectly logical because sticking to one system maximizes the number of ways that pieces can fit with one another. However, the possibility of combining more than one angle system can also lead to valid puzzles. The purpose of this article is to show how puzzles might be built using pieces from different angle systems along with pieces that are a combination of more than one system. Hopefully this opens up a new world of possibilities to be explored!


The Pieces

Spheres can be arranged at their tightest into an “iso-” system, a 60° grid which is the same pattern that hexagons make. Alternately, spheres can be arranged more loosely into an “ortho-” system, the 90° grid approximating a pattern of squares. Within each of these grids we can look at the possibilities of how three or four spheres might be attached together in order to make different puzzle pieces. Using a 60° system, spheres can be used to approximate the 3 trihexes (pieces A, B, and C ) and the 7 tetrahexes (pieces F, G, H, I, J, K and BB). Using the 90° system, we find that the pieces approximate the 2 triominoes (pieces C and D) and the 5 tetrominoes (pieces L, M, N, O, and BB).

Table of Pieces


The new pieces are mixtures of the 60° and 90° systems. Piece P, for example, has both a 60° angle between three of the spheres and a 90° angle between a different three. Often the mixture of the two systems can be seen in a 150° angle (60° + 90° ). This is exemplified by the three-sphere piece E. It is the only possible mixed-angle piece built of three spheres. The 12 pieces labelled “P” to “AA” contain all of the twelve possibilities for how 4 spheres can be combined within the 60° /90° mixed system.  As far as I know, these “tetra-mix” pieces are here presented for the first time. Their shapes are new to the puzzle connoisseur, despite the fact that they consist of combinations of only 3 or 4 elements. Technically labelled “ortho-iso-tetraspheres,” I call them tetra-mixes for short. These pieces will not fit into either the 60° system or the 90° system, but will only fit into a pattern that alternates between the two systems because they represent border patterns that appear only at points of juncture between the two systems.


We now have a combined system of 28 possible puzzle pieces from which we could draw to make a virtually unlimited number of puzzles. The possibilities include 5 “tri” (Pieces A,B,C,D, and E) and 23 “tetra” pieces. Of the pieces, 8 belong solely to the 60° system, 5 belong solely to the 90° system, 13 belong to the mixed 60/90° system and 2 (Pieces C and BB) could belong to either the 60° or the 90° systems, but are not in themselves a mixture of both. Using all 28 of the possibilities would make for a rather bulky puzzle and one that would need to have a distinctive and unusual shape, but many satisfactory puzzles can be made from subsets of the pieces.


We might notice that there are twelve of the tetra-mix pieces. This is somewhat an ideal number in puzzle history as seen before with the 12 pentominoes (all the possibilities of combining five squares) or 12 hexiamonds (all the possibilities of combining six equilateral triangles). However, Piece AA is for the most part unusable because it has an inner “pocket” that will inevitably leave a third, even larger, size hole—an irregular five-pointed “star.” All puzzles made using the combined system will have at least two sizes of holes, and keeping it down to only two sizes (the three and four-pointed stars) makes for better puzzles, so we will avoid using this piece. If we substitute the single tri-mix piece, however, we still have a total of 12 useful “mix” pieces. For example, Puzzles 12, 13, and 14 use these twelve “mix” pieces.



Designing Puzzles


Spheres must be used (rather than hexagons, for example) since all building blocks of the puzzle must have the possibility of being part of either a 90° or a 60° system (individual spheres are usually part of both). The disadvantage to using spheres is two-fold: The first problem is that the point of juncture between the spheres is conceptually a point, rather than a line or edge, which can lead to weak puzzle pieces. But a bit of fudging to enlarge the contact area and reinforce the juncture points has been successfully accomplished in previous puzzles. The second problem is somewhat more serious: unlike squares or hexagons, circles leave holes between the pieces which cannot be filled in. In a pure 60° system, this is not a big problem (even if conceptually messy) because all the holes will be of the same size. In a mixed system, however, the holes will be of more than one size. For example, the holes in the 60° system form a three-pointed “star” while the holes in the 90° system are larger and shape a four-pointed “star.” The size of holes allowed within any puzzle and precisely where each size occurs can be very confusing to the puzzle solver. This we accept as part of the problem involved in solving the puzzle, but it also puts a practical limit on the size of puzzles, because the number of possible hole patterns (which actually reflect the 60° / 90° system interactions) becomes too large and overwhelming for the puzzle solver as the size of the puzzle increases. This problem too can be avoided by giving the entire pattern rather than just a frame for the pieces.


What criteria should be used in designing the puzzles? Historically for these types of puzzles, the greater the amount of symmetry, the more pleasing the shape is considered. All of the puzzles presented here have at least bilateral or trifold symmetry. Another criteria is to try to “round out ” the outer edge. Though interesting shapes may result by using sharp projections, these often make a puzzle much easier because only a certain few of the pieces can fit into them. In particular, single spheres along the edge that only attach to one other sphere (as in Puzzle 5, for example) should be generally avoided. Another aim sometimes cited by puzzle designers is to produce the most difficult puzzle with the fewest number of pieces. In keeping with this tradition, most of the puzzles presented here have only seven or eight pieces, but they could easily be built up in various ways to produce larger puzzles. The “difficulty level” is hard to predict without a working model in hand that can be tried out by many people—and then opinions can vary widely. One final goal will be to use as many of the “mix”pieces as possible and as few of the pure 60° or 90° pieces as necessary.


One way of finding new patterns for puzzles for these pieces is to use various germ cells and build around them. Starting with the simplest, a single sphere, we can build a ring of spheres around it to give the pattern in Example 1. This pattern belongs entirely to the 60° system, and so it is important that we try to introduce the 90° system in the next layer if we want to have a puzzle that might be solved with only tetra-mix pieces. The best way to assure this is to choose two of the outer spheres and add two more spheres at right angles in order to make a pattern into which Piece O, the “square” of spheres, could fit (Example 2). Continuing this pattern around we arrive at the configuration for Puzzle 1. We could continue adding layers in the same manner or we can begin to add spheres individually, as for example the six “points” of the star in Puzzle 3.

In a similar fashion we could build up layers around cells of two, three or four spheres (Examples 3-6). The single sphere is the germ for Puzzles 1-14, while the double sphere resides at the center of Puzzles 15-24. Puzzles 25-40 use a three-sphere cell, while Puzzles 41-54 use a rhomboid center. Puzzles 55-63 use a “square” center while Puzzles 64 and 65 use a five-sphere cross (or nine-sphere square). Puzzles 66 and 67 are centered around 2 three-sphere cells in mirror image.

To see how puzzles can be built, let’s take for an example the three-sphere cell (or Example 4). Rather than a central group of three spheres with a ring around them, Example 4 could be seen as a group of three rhomboids (Example 7) or as three interlocked “squares” with three single spheres added (Example 8). These examples elucidate two ways that one may go about building puzzles: by adding “squares” and filling in or by using building blocks of 60° pieces fit together at 90° angles. For example, we can add two more spheres in three places around the edge of Example 4 to make three different “squares” as seen in Example 9. Filling in the rest of the layer with more spheres gives Puzzle 32. On the other hand, the same puzzle could be arrived at by piecing together groups of three and five spheres as seen in Example 10.

Starting again with Example 4 as a base, adding one sphere in three spots gives the somewhat unwieldy Puzzle 25 or adding a group of three spheres along three sides gives Puzzle 26. Adding a rhomboid along three edges gives Puzzle 27 and adding two more spheres to each side of that gives Puzzle 36. Incidently, Puzzles 25, 27, and 36 have trifold symmetry, but no bilateral symmetry. A third layer of spheres could be built around Example 5 in two different ways to form either Puzzle 32 or 33. Individual spheres could be plucked from these outer layers to give many different puzzles, for example Puzzle 28. Or these two outer layers could be alternated with each other to give puzzles like Puzzles 30, 31, 34, and 35, or the triple alternation of Puzzle 29. Returning to Puzzles 32 and 33, we could now add individual spheres to open up many new possibilitites with Puzzle 37 as an example. We could then continue the process adding a fourth layer for Puzzle 39 (one of two possibilities) and begin taking away spheres again as in Puzzle 38. Using another approach, six “squares” are added to Puzzle 32 with one of the spheres from each square overlapping one of the spheres from the outer layer of Puzzle 32 to give Puzzle 40.


The Puzzles

One of the best puzzles is the “star” or “snowflake” puzzle (Puzzle 3). It is made of only 7 pieces: 3 pieces each built of three spheres and 4 pieces each built of four spheres. Piece A is entirely within the 60° system , while Piece D lies entirely within the 90° system. As simple versions of the two systems, these pieces make it immediately clear to the puzzle solver that these two different systems are being used. Piece E is 150° (90° + 60° ), a combination of the two systems, and so the three systems are represented by one “tri” piece each. All of the rest of the pieces are tetra-mixes. This design has the highest degree of symmetry possible with these pieces—six-fold (60° rotational ) symmetry as well as bilateral symmetry along three axes.

Isn’t it, however, a bit complicated to ask the puzzle solver to understand how the 90° and 60° systems are enmeshed to make this puzzle if only given a frame to set the pieces in? Yes, it is. But a thorough understanding of the mixed system behind the puzzle is not at all necessary to solve it. The shape of the tray for Puzzle 3 (combined with the shape of the pieces) inevitably leads to a correct pattern in placement of the pieces. I have had eight of my friends (admittedly not a big sample—but some of them profess to be extremely bad at puzzles) try out Puzzle 3. Only about half had the patience to continue until they solved it, but all of them rather quickly came to a point where they were laying the pieces into the correct pattern. They couldn’t solve it because they kept coming up with one or two pieces at the end that didn’t fit the space that was left. For me, this is one of the signs of a good puzzle. Example 11 is an example of a typically frustrating ending point often reached when trying to solve this puzzle. Here we are left to place the 90° Piece D, but are instead left with a hole appropriate to the 60° Piece A. The puzzle seems impossible, but the solution shown by Puzzle 3 proves that it is not. In fact, there are eighteen different solutions to the puzzle (not counting rotations and inversions—then there would be 216!), but most solvers are lucky if they can even find one! At least three other combinations of seven pieces from the original 28 can be used to solve Puzzle 3. Can you find more?

Puzzle 4 is another version of the star puzzle. It differs in that the outer “points” of the star are rotated 30° from Puzzle 3. Or it could be seen as a 30° rotation of the seven inner spheres of Puzzle 3. Can you find a solution using pieces B, D, E, R, T, W, and Y?

Puzzle 1 is in the shape of a circle built with just five pieces—A, P, Q, U, and Y. Another version uses Pieces D, P, U, X, and Y. If these are too easy a challenge, can you find a solution that does not include Piece P? How many different combinations of five pieces will solve the puzzle? (answer unknown)

The answer is given for Puzzle 3, can you find answers for the rest? Remember to try to use the most “mix” pieces possible, but not all puzzles can be solved with only “mix” pieces. Contrarily, can you solve any with only 60° or 90° pieces and without any mix pieces or duplicate pieces? (answer unknown)

Good puzzling!

Copyright © 1999 by William Waite, all rights reserved.

Leave a Comment

Your email address will not be published. Required fields are marked *