§1. The Smart Games company has developed a tile-fitting puzzle named IQ Hexagon (briefly, "IQH"). We examine some of its mathematical aspects.

Step A of figure 1, shown below, introduces the puzzle by displaying one of the many solutions possible with the factory-supplied tiles (twelve in number), and the board on which they are placed. This particular solution appears in the factory's instruction book. For clarity in the images below, some changes have been made:

• the tile colors, which are not significant in play, have been altered in order to more clearly distinguish adjacent tiles;
• the corners and ends of the tiles have been reshaped from round to polygonal, to better illustrate how they fit one another;
• lightened from black to gray are the border and the studs, where tiles are not to be placed.

figure 1, step A (larger image)

The remaining steps will ultimately show how each tile corresponds to a well-researched mathematical object, namely the polyhex.

• Step B adds letters and numbers for reference; the letters (assigned by the factory) designate the tile, and the numbers (devised for this report) indicate regions of the tile. Meanwhile, studs are marked with an "x".
• Step C simplifies the coloring.

figure 1, step B (larger image)	figure 1, step C (larger image)

• Step D draws lines, dividing the tiles into hexagons and triangles.
• Step E removes the triangles.

figure 1, step D (larger image)	figure 1, step E (larger image)

• Step F rotates all the hexagons by 30 degrees, and expands them to fill the gaps.
• Step G removes the studs, and then bridges the hexagons of each tile, revealing polyhexes.

figure 1, step F (larger image)	figure 1, step G (larger image)

The size of a tile is the number of hexagons in its corresponding polyhex. The twelve factory-supplied tiles are:

• size 5: I and K
• size 7: E and G
• size 6: all others

Many other shapes of tile in this style are possible, and it is not obvious why the factory chose these twelve in particular. An expansion kit containing other shapes of tile would make perfect sense from a puzzle-solving point of view.

There is a definite correspondence between polyhexes and IQH's tiles, and this can be extended to the many additional tiles of the same style. However, it is not a one-to-one correspondence, and the discussion below examines that.

We use different notations for the factory tiles (as above) and the tiles invented for this report, which will begin appearing in §2:

• factory tiles: each tile has an uppercase letter, and each region within the tile has a different number (for example E2);
• invented tiles: each tile has a number, and each region of the tile has a different lowercase letter (for example 32f).

§2. All the factory tiles except D have twelve orientations; tile D, with an axis of symmetry, has only six. Inversion occurs when the tile is physically picked up and flipped over. To give an exmple, figure 2a shows all twelve orientations of tile B:

figure 2a tile B	rotations
inversions

Non-factory tiles can have various numbers of orientations, for example:

	figure 2b
	1 orientation		2 orientations	3 orientations	4 orientations	6 orientations		12 orientations
	tile 21	tile 22	tile 23	tile 24	tile 25	tile 26	tile 27	tile 28
step A
step D

Although tiles 26 and 27 each have six orientations, there is a substantive difference:

• Tile 26 can achieve all six of its orientations without being flipped over; so no new orientations result if it is flipped over.
• Tile 27 can achieve only three orientations without being flipped over; but if it is, three other orientations result.

If flipping a tile over yields new orientations, it is said to be chiral. Thus tile 27 is chiral, and 26 is not. All the factory tiles are chiral except tile D.

The divisors of the number 12 are 1, 2, 3, 4, 6, and 12 itself. There do exist IQH tiles with each of those numbers of orientations, as in figure 2b. Also, no tile in the style of IQH will have any other number of orientations.

§3. In some cases, several different tiles are equivalent for solving the puzzle; this happens precisely when they resolve to the same polyhex. Figure 3a displays four such tiles. Tile 31 is simply a rotation of tile F in figure 1, while tiles 32 and 33 were invented in a style typical of the factory's tiles. Tile 34, although equivalent to the others, has a large triangular region not found in any of the factory's tiles.

Steps A through G of figure 3a correspond to the steps of figure 1. The Step D extract highlights the fundamental differences of the tiles: different acute corners are filled in.

figure 3a
	tile 31	tile 32	tile 33	tile 34
step A
step B
step C
step D
step D extract
	all four tiles are the same in the remaining steps
step E
step F
step G

As seen in the step D extract, each of tiles 31, 32, and 33 has two corners filled in, while tile 34 has all three. Because tile 34 is thus distinguished from the other three, we recommend that this treatment be declared canonical. Not every polyhex has a valid counterpart as a tile in the IQH puzzle; this is because of the placement of studs on the playing board. But if a polyhex does have a corresponding canonical tile, there is only one.

Figure 3b shows the effect of canonizing (short for "canonicalizing") all the factory tiles. Opinions will vary as to whether this is an improvement, but in any case the underlying polyhex structure is less conspicuous.

figure 3b
original factory tiles same as figure 1, step A	tile 35 canonized factory tiles

Finally, the puzzle would work the same if all the tiles were manufactured in precisely the shape of their polyhex equivalent, as in step G of figure 1.

There are no particular constraints on the size or shape of the board. Figure 4 contains a few examples of what can be done.

Recall that each tile can be converted into a polyhex, as in the sequence of figure 1. When a tile is correctly placed into any of the boards below, each of the polyhex's component hexagons will fit onto one of the numbered pink hexagons.

The numbering of the pink hexagons demonstrates what total size of tiles will be necessary for a solution. Note that, from one board size/shape to another, the various totals do not have a common factor. For instance, figure 4e contains 52 = 4 × 13 hexagons, while 4f contains 69 = 3 × 23.

The gray hexagons are the usual studs, as in figure 1.

figure 4a — original	figure 4b	figure 4d
	figure 4c — minimal

The overall shape need not be a regular hexgon:

figure 4e	figure 4f	figure 4g

There can be a hole in the middle. Figure 4i is what is taken out of 4d to produce 4h:

figure 4h	figure 4i