§1: Introduction. 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

The remaining steps will ultimately show how each tile corresponds to a well-researched mathematical object, namely the polyhex. Polyhexes may be categorized in different ways, but IQH relies on the kind termed free, where "rotations and reflections count as the same shape" [Wikipedia].

• 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 "s".
• Step C simplifies the coloring.

figure 1 — step B	figure 1 — step C

• Step D draws lines, dividing the tiles into hexagons and triangles.
• Step E removes the triangles, thus clearly disclosing hexagons. A hexagon where a tile may be placed, in other words a non-stud, is termed tilable (although some people may prefer the longer spelling tileable). Studs are rendered in gray, and tilable hexagons are not.

figure 1 — step D	figure 1 — step E

• 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	figure 1 — step G

For this particular solution, three colors were enough to ensure that adjacent tiles were of different colors. There may be solutions where four colors are required, but never more than four.

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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

The component hexagons total 72, which of course is the number of tilable locations on the board.

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 remaining discussion examines that.

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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's name is an uppercase letter in the range A-L. This is followed by a number for each region within the tile (example E2);
• Invented tiles: Each tile's name begins with a numeral, indicating in what section of this report it is introduced. Further characters vary according to the context.

§2: Orientations. All the factory tiles except D have twelve orientations; tile D, with an axis of symmetry, has only six.

When a tile is physically picked up and flipped over, the action is called inversion, and it can be responsible for generating additional orientations. For IQH tiles and polyhexes, inversion has the same effect as mirror-image reflection. If inverting a tile does yield new orientations, the tile is said to be chiral; otherwise achiral. All the factory tiles are chiral except tile D.

To give an example, figure 2a shows all twelve orientations of tile B:

figure 2a: tile B, step A	rotations
inversions

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

		figure 2b
		step A		step D		step F		chiral?
1 orientation	tiles 21 and 22							no
2 orientations	tiles 23 and 24							no
	tile 25							yes
3 orientations	tile 26							no
4 orientations	tile 27							yes
6 orientations	tiles 28 and 29							no
	tile 2Z							yes
12 orientations	tiles 2Y and 2X							yes

In the table above, the number of orientations applies to the physical form of the tile (step A); the corresponding polyhex (step F) might have fewer. Tile 2X is an example. Also, observe the difference between tiles 27 and 2X: although they agree in step F, they differ in step A. Such equivalent tiles are examined in the next section.

A tile with:

• 1 or 3 orientations is achiral;
• 4 or 12 orientations is chiral;
• 2 or 6 orientations might be either.

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. Neither any polyhex, nor any tile in the style of IQH, will have any other number of orientations.

§3: Equivalent tiles. In some cases, several different tiles are equivalent for solving the puzzle — they may be freely substituted one for another. This happens precisely when they correspond 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. Also invented is tile 34 which, 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 F	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. As discussed in §4 below, not every polyhex has a valid correspondent 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	canonized factory tiles

As shown in figure 3c below, factory tile G has two acute corners and no symmetry, and as a result it has 4 × 4 = 16 variations. This multiplicity makes a strong case for defining one version as canonical, in order to assist in the management of shapes. In the figure below, version 1a is the factory original. Row 4 is canonical in the angle on the left (G3), and column D is canonical in the angle on the right (G6). Hence version 4D is fully canonical.

	figure 3c — variations on factory tile G
	A	B	C	D
1
2
3
4

A tile with three acute angles and no symmetry would have 4 × 4 × 4 = 64 variations, for example tile 35 in figure 3d:

figure 3d
fully uncanonized	fully canonized
		There are: 26 other fully-uncanonized versions; 27 once-canonized; 9 twice-canonized; 1 fully-canonized (as left).

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

§4: Eligibility of polyhexes.

§4A: Ineligible. Some candidate tiles in the IQH style will not work, because a stud would get in the way. These can be detected by examining their polyhex correspondents. In figure 4a are three examples. Sure to fail is any polyhex that contains as a sub-polyhex the wye shape of tile 41, the diamond shape of tile 42, or the offset shape of tile 43.

It happens that the tetrahexes (seven in all) have been given standard names; three of them are as shown below.

	figure 4a
	step A	step B	step C	step D	step E	step F	name
tile 41 will NOT work							"propeller"
tile 42 will NOT work							"bee"
tile 43 will NOT work							"wave"

Figure 4b displays tile 44, which will work. By contrast, tile 45 will not work, even though it differs only slightly from tile 44. The reason: the middle section of tile 44 has an odd number of hexagons (44b-c-d), while the middle section of tile 45 has an even number (45b-c-d-e). This extends to other odd and even numbers.

	figure 4b
	step A	step B	step C	step D	step E	step F
tile 44 will work
tile 45 will NOT work

Figure 4c modifies tiles 44 and 45, producing tiles 46 and 47 respectively. The one difference is that the lower leg is bent in the other direction, but this changes whether the tile will work. The odd-and-even extension rule of figure 4b still applies, but in reverse.

	figure 4c
	step A	step B	step C	step D	step E	step F
tile 46 will NOT work
tile 47 will work

Comparing tile 44 with tile 46, or tile 45 with tile 47, it can be seen that altering one angle changes the tile between working and not working. For some tiles, however, changing an angle does not affect whether it will work. An example of this is found with tiles 28 and 29, shown in figure 2B above: both tiles work.

Key Observation • Each IQH-style tile corresponds to exactly one polyhex. • Each polyhex might correspond to no IQH-style tile, exactly one tile, or a larger number.

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§4B: Eligible. Wherever a valid tile has two adjacent hexagons, the line defined by their centers can be extended arbitrarily far in both directions to produce a larger valid tile. Figure 4d offers an example based on factory tile L1 and various parts of invented tile 48:

	figure 4d
tile L → tile 48 will work
	original	extending L1-L2	extending L3-L4	extending L5-L6 (which incorporates L2-L3)	extending all

Tile pairs L4-L5, L1-48k, and 48d-48f can generate further extensions. This procedure is greatly extended in §5 below.

Here are two examples of extensions from tiles defined earlier:

	figure 4e extract from figure 2b
tile 22 → tile 25 and tile 23 → tile 27 will work

Going the other way, removing hexagons from a valid tile results in a valid tile, if the remaining hexagons remain connected:

	figure 4f extract from figure 2b
tile 24 → tile 28 will work

Adding or removing hexagons will often change the number of a tile's orientations.

§5: Sufficiency. The method of figure 4d, applied enough times, is sufficient to fill the entire board. From that, the method of figure 4f can be used to remove any undesired tiles. Together, they allow any valid tile to be constructed. We have produced an example of covering the standard IQH board in eighteen steps, but because of its length it is detailed on a page of its own. However, as a glimpse into the procedure, here are the start and finish:

figure 5a — start	figure 5e — finish
details here

§6: Board sizes and shapes. There are no particular constraints on the size or shape of the board. Figures 6a-b-c-d contain a few examples of what can be done by extending or reducing the factory pattern of studs and tilable hexagons.

Recall that each tile can be converted into a polyhex, as in the sequence of figure 1. When a valid tile is correctly placed into any of the boards below, each of the polyhex's component hexagons will fit onto one of the numbered green tilables. Meanwhile, the gray hexagons are the usual studs, as in figure 1.

The numbering of the tilables shows 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 divisor. For instance, figure 6b1 contains 69 = 3 × 23 hexagons, while figure 6b3 contains 74 = 2 × 37 hexagons. There is no simple general formula for calculating the number of hexagons in a board, but there is a convenient formula that will handle many cases.

Figure 6a displays several regular hexagons.

figure 6a1 — minimal	figure 6a3 — factory	figure 6a4
figure 6a2

The overall shape need not be a regular hexagon. In particular, the shape of figure 6b3 may be convenient for a player who is resting the board on their lap.

figure 6b1	figure 6b2	figure 6b3

Figures 6c1 and 6c2 are the same approximate rhombus, except differing in their treatment of the acute corners. Figures 6c3 and 6c4 are derived from figure 6a4 by removing sections, revealing that the board can have a hole, and that it need not be convex.

figure 6c1	figure 6c3	figure 6c4
figure 6c2

Figure 6d1 shows that, except for edge effects, the number of tilable hexagons is three times the number of studs. Each of the characters "0", "1", "2", and "3" appears 37 times, in one-to-one correspondences:

• each "0" has a "1" to its northwest, and each "1" has a "0" to its southeast;
• each "0" has a "2" to its southwest, and each "2" has a "0" to its northeast;
• each "0" has a "3" to its east, and each "3" has a "0" to its west.

figure 6d1	figure 6d2

Figure 6d2 demonstrates that along the periphery of the board, the hexagons that have been removed (red with X) alternate with those that remain (green with number).

§7: Tessellation.

Figure 7a is based on the factory version of IQH, except that the board is not drawn, and some of the tilable hexagons along the edge have been removed to prepare it for tessellation. However, all nineteen studs remain. It corresponds to figure 6d1, differing only in size.

Figure 7b demonstrates the tessellability of figure 7a, with adjacent instances in different colors. If this were the ultimate extent of the board, the gaps at the edges would presumably be filled in.

figure 7a

figure 7b

The board in figure 6d1 is tessellable in the same manner as 7a into 7b, but requires an unwieldily large image to convey that fact.

Below, figure 7c shows the pattern of figure 6a1, which can be tessellated (figure 7d) without removing any of the edge hexagons. Although 7c is achiral (except for the numbers), its tessellation in 7d is chiral.

figure 7c

figure 7d

§8: Non-factory stud patterns. Boards with the factory pattern of studs have a ratio of three tilable hexagons for every stud, except at the edges, as pointed out in the discussion of figure 6d1. However, there are many other possibilities, often related by transforming one pattern into another.

Examples are given below, where most images contain an arbitrary, but convenient rectangular subset of a pattern that can be extended as far as desired. The studs are numbered when it will aid understanding.

§8A: Expansion and compression. Figure 8a1 has a ratio of two tilables per stud, which is the minimum ratio in an arrangement where no studs are adjacent. Figures 8a2 and 8a3 result from adding tilable hexagons to expand the figure horizontally. Expansion and its opposite, compression, can be applied to many patterns.

figure 8a1 — ratio 2:1	figure 8a2 — ratio 4:1	figure 8a3 — ratio 6:1

§8B: Shifting. The patterns of figures 8b1 and 8b2 are like that of 8a3, except that rows have been shifted:

• in 8a3, each stud is directly under tilables numbered 3 and 4;
• in 8b1, each stud is under 2 and 3;
• in 8b2, each stud is under 1 and 2.

One more shift would cause the studs to be adjacent in long lines. Although valid, this would lead to a puzzle that many people might find uninteresting.

Shifting so that each stud is under 4 and 5 would give the reflection of 8b1; under 5 and 6, 8b2.

figure 8b1 — ratio 6:1	figure 8b2 — ratio 6:1

§8C: Substitution.

• The pattern of figure 8c1 is the same as that of 8a1, although shown with more horizontal iterations. Studs 21 through 32 form a rectangular grid of their own, which, although smaller, is of the same nature as the grid of studs 1 through 20.
• If studs 21 through 32 are replaced with tilables, figure 8c2 results, with a 5:1 ratio of tilables to studs.
• Figure 8c2 may then be compressed to produce figure 8c3 which has a 3:1 ratio, but which is not the factory pattern. However, the factory pattern can be obtained by shifting studs 5-6-7-8 and 13-14-15-16 by one hexagon horizontally.

figure 8c1 — ratio 2:1	figure 8c2 — ratio 5:1	figure 8c3 — ratio 3:1

§8D: Multiple adjacent studs. This is not a transformation method, but rather the lifting of a restriction. The pattern shown in figure 8d is best explained by showing a tessellable unit (8d1) and a sample board (8d2). Figure 8d3 is like 8d2, except that the tilables and studs are numbered straight through.

figure 8d1 — ratio 18:7 ≈ 2.571

figure 8d2 figure 8d3

In general, patterns that use multiple adjacent studs will need to be large in order to make the repetition of the pattern obvious. Of course, the transformation methods of §8A-B-C can also be applied here.

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§4 above shows several tiles (41-47) that will, or will not, work in the factory stud arrangement; and tile 48 is used to illustrate an extension procedure that will work. However, attempting to place any of these tiles on non-factory stud arrangements may yield different degress of success.

Widely available are polyhex-based puzzles with no studs or other obstructions at all, and they typically include tiles whose polyhex structure is explicit. In fact, on the internet are easily found patterns suitable for three-dimensional printers. However, the presence of studs in the IQ Hexagon puzzle lifts it out of the run-of-the-mill, and renders it a much more nuanced challenge. Indeed, we would have had no reason to write this report if the studs had been omitted.

§9: Miscellaneous.

This page was created on a Macintosh computer. We used two components of Apple's Xcode: the text editor for preparing the HTML, and the C++ compiler running our own code for producing the Scalable Vector Graphics (SVG) images.

Because the images are in a vector format rather than a raster format, they can be enlarged or reduced with negligible loss of accuracy. Each of the over 200 images is in a separate file, which the general public is welcome to download, modify, and republish.

The somewhat awkward letter-and-numbering scheme for invented tiles reflects two factors:

• there is room for only three characters for each label within the images;
• it would be confusing to have, for instance, a tile containing the label "2a" or "2A" when there is already a figure named "2a".

This results in the peculiar sequence of tile numbers 51-52-53-54-55-56-57-58-59-5Z-5Y-5X-5W-5V-5U-5T-5S-5R of the §5 annex.

The coordinate system for hexagons used internally within the C++ program employs three numbers (named a, b, and c) summing to zero. The redundancy provides more symmetry than ordinary x, y coordinates would, and the symmetry in turn provides convenience and better error detection. Most hexagons in this report have coordinates that are multiples of three so that the few exceptions will still be integers. The coordinates look like this:

figure 9a

As a digression, here is the HTML for some Unicode hexagons:

name	char	code	char	code
WHITE HEXAGON	⬡	&#x2B21;	⬡	<span style="color:#FF0000";>&#x2B21;</span>
BLACK HEXAGON	⬢	&#x2B22;	⬢	<span style="color:#00FF00";>&#x2B22;</span>
HORIZONTAL BLACK HEXAGON	⬣	&#x2B23;	⬣	<span style="color:#0000FF";>&#x2B23;</span>
HORIZONTAL WHITE HEXAGON	not found