Serpentiles

An orthogonal rotation that leaves the hexagon in the same orientation (with flat sides to the left and right) is 60°. There are six potential orthogonal rotational positions (360°/60°) of the regular six-sided hexagon. By examination, 300 has rotational symmetry: any rotation of 60° will result in the same sides being linked, so 0300 has one unique rotational position. Similarly, 003 has two unique rotational positions; one rotation of 60° will result in different sides being linked by the paths, but a second rotation of 60° in the same direction will result in the original sides being linked again. In a similar fashion, 102 and 120 both have three unique rotational positions, and 021 has six unique rotational positions.

Orthogonal rotations of 003

Orientation 12-34-56

Orientation 23-45-61, after one orthogonal rotation

To uniquely describe each rotational position, imagine a reference frame is applied where each side is numbered from 1 to 6 sequentially anti-clockwise. The paths connecting sides can be described as linked pairs.

For example, for 003 above, proceeding anti-clockwise from the lower right edge, the path/pair notation is 12-34-56. A single orthogonal rotation of the shape will add one (if anti-clockwise) or subtract one (if clockwise) to each number, modulo 6. In this way, the linked paths with an orthogonal rotation (60°) anti-clockwise would change to 23-45-67, or rewriting using modulo 6, 23-45-61, demonstrating that different sides are linked with a single orthogonal rotation. The specific order within a pair and the order in which the pairs are written does not affect the configuration. For example, 23-45-61 is equivalent to 23-45-16 and 16-23-45. Thus with a second orthogonal rotation, which results in 34-56-12, it is simple to demonstrate the second orthogonal rotation is equivalent to 12-34-56, the original orientation.

The reference frame could also be numbered sequentially in a clockwise fashion. In that case, a clockwise rotation would increment the values by 1, while an anti-clockwise rotation would decrement the values by 1. The orientation of the reference frame (clockwise or anti-clockwise) only affects the arithmetic required to rotate the tile. Likewise, the choice of origin (which side is designated the first side) for the reference frame is arbitrary, but the reference frame remains fixed during rotation.

Describing the tiles by linked sides also implicitly includes the three-digit Van Ness notation. For any arbitrary three-pair sequence AB-CD-EF, the Van Ness notation can be recovered by the following formula: ${\displaystyle 10^{[min(\left\vert B-A\right\vert ,6-\left\vert B-A\right\vert )-1]{\bmod {3}}}+10^{[min(\left\vert D-C\right\vert ,6-\left\vert D-C\right\vert )-1]{\bmod {3}}}+10^{[min(\left\vert F-E\right\vert ,6-\left\vert F-E\right\vert )-1]{\bmod {3}}}}$

Although a regular six-sided polygon (hexagon) can be used to cover a plane surface without gaps, a regular four-sided polygon (square) also can be used. The square with two paths linking sides can be described in a similar fashion, using a two-digit notation yz; there are just two potential configurations.[3]

• 4S2P: Four-sided polygons with two paths
• 
  02, indicating the paths link two adjacent sides
• 
  20, indicating the paths link two opposite sides

Like the hexagonal case, 20 (where all paths link opposite sides) has rotational symmetry. It is clear that 02 does not have rotational symmetry, as rotating the tile 90° clockwise or anti-clockwise will link different sides. Compare with the same tile rotated by 90°: Square tiles that lack rotational symmetry are known as Truchet tiles, as described in a 1704 memoir by Sébastien Truchet entitled "Mémoire sur les combinaisons" and popularized in 1987 by Cyril Stanley Smith.[4][5]

Likewise, a regular eight-sided polygon (octagon) can have four paths pairing the eight sides, described using Van Ness's notation in a similar way with a four digit notation wxyz. There are 18 distinct combinations:[3]

• 8S4P: Eight-sided polygons with four paths
• 
  0004
• 
  0022
• 
  0040
• 
  0103
• 
  0121(a)
• 
  0121(b)
• 
  0202
• 
  0220
• 
  0301
• 
  0400
• 
  1012
• 
  1111(a)
• 
  1111(b)
• 
  1210
• 
  2002
• 
  2020
• 
  2200
• 
  4000

As with the preceding hexagon and square examples, the case linking all the opposite sides (4000) has rotational symmetry. Note that for the octagon, the notation no longer uniquely describes the path configuration. There are two duplicate notations in the gallery: 0121 and 1111. The 1111 pair are mirror-images. The rotational position notation can be applied to distinguish these tiles; with a reference frame numbered anti-clockwise starting from the bottom edge, 0121(a) above can be written as "0121 12-36-47-68", which is distinct from 0121(b): "0121 12-38-46-57". Likewise, 1111(a) is "1111 12-36-48-57", distinct from 1111(b): "1111 13-25-48-67". Using modular arithmetic (modulo 8), 1111(a) and (b) can each be rotated by one orthogonal rotation (45°) anti-clockwise to positions "15-24-36-78" and "15-68-47-23", respectively, demonstrating their paired reflective symmetry.

These octagonal tiles could be combined with the square tiles described here (rotated through 45° to fill the interstitial spaces) to completely tile a surface with paths across all edges.[6]

The preceding discussion covered the cases of regular 4-, 6-, and 8-sided polygons with a single entry point per side, with each side linked by a path. Because the paths link two different sides, using a single entry point per side requires regular 2n-sided polygons with an even number of sides to ensure that all entry points are linked by paths. If there are tiles with an odd number of sides, an even number of entry points would similarly ensure that all entry points are linked by paths.

• 3S3P: Three-sided polygons with three paths
• 
  003(a) 12-34-56
• 
  003(b) 16-23-45
• 
  021(a) 12-35-46
• 
  021(b) 13-26-45
• 
  102 12-36-45
• 
  120 15-24-36
• 
  300 14-25-36

Both 003(a) and (b) are reproduced here; the (b) variant is recovered by the inscribed triangle.

For example, a regular 3-sided polygon (equilateral triangle) could have 2 entry points per side. There are 7 potential combinations using 3 paths per tile, depicted above.[3] Although the Van Ness notation can be applied by modifying it to count adjacency rather than sides, each tile is not uniquely described by the Van Ness notation, although the relationship between the 003(a) and (b) cases can be seen by inscribing a triangle, as the example shown here. A similar triangle could be inscribed for the 021(a) case to demonstrate its relationship to 021(b).

It may be simpler to describe the tile by path pairings instead. In the set of three-sided tiles with two entry points per side and three paths, the numbering starts from the left-hand point on the bottom edge and proceeds sequentially anti-clockwise.

One complete rotation turns the tile by 120° (=360°/3 sides). Rotation arithmetic is performed by adding two, modulo six. Consider the tile (12-34-56). Rotating the tile one or more complete rotations will, by inspection, not change the points that are linked. To demonstrate this, one rotation anti-clockwise results in the configuration (34-56-78), or, after the modulo arithmetic is performed, (34-56-12), which can be rearranged to (12-34-56), showing that one rotation does not affect the linked paths and so there is one distinct rotational position for this tile. On the other hand, consider the tile (13-26-45). The first two anti-clockwise rotations of this tile result in the configurations (16-24-35) and (15-23-46), respectively, so this tile has three distinct rotational positions.

Similarly, a regular 4-sided polygon (square) with 2 entry points per side has 35 potential combinations using 4 paths per tile.[3] The complete rotation is 90° and the rotational symmetry can be determined by adding 2, modulo 8 in a manner similar to that for the equilateral triangle case above. This set of four-sided polygons with two entry points per side and four paths per tile is used in the commercial board game Tsuro.

• 4S4P: Four-sided polygons with four paths
• 
  0004(a) 12-34-56-78
• 
  0004(b) 18-23-45-67
• 
  0022(a) 12-34-57-68
• 
  0022(b) 18-23-46-57
• 
  0040(a) 17-24-35-68
• 
  0040(b) 17-28-35-46
• 
  0103(a) 12-34-58-67
• 
  0103(b) 12-38-45-67
• 
  0121(a) 12-35-47-68
• 
  0121(b) 17-23-46-58
• 
  0121(x) 12-38-46-57
• 
  0121(y) 17-28-36-45
• 
  0202(a) 12-38-47-56
• 
  0202(b) 18-27-36-45
• 
  0220(a) 16-28-35-47
• 
  0220(b) 17-24-36-58
• 
  0301(a) 12-36-47-58
• 
  0301(b) 16-23-47-58
• 
  0400(a) 14-27-36-58
• 
  0400(b) 16-25-38-47
• 
  1012(a) 12-35-48-67
• 
  1012(b) 12-37-45-68
• 
  1111(a) 16-28-37-45
• 
  1111(b) 13-25-48-67
• 
  1111(x) 12-36-48-57
• 
  1111(y) 12-37-46-58
• 
  1210(a) 13-26-47-58
• 
  1210(b) 15-28-36-47
• 
  2002(a) 12-37-48-56
• 
  2002(b) 18-26-37-45
• 
  2020(a) 13-26-48-57
• 
  2020(b) 15-28-37-46
• 
  2200(a) 16-25-37-48
• 
  2200(b) 15-27-36-48
• 
  4000 15-26-37-48

Inscribed relationships

0400(a), inscribed 0400(b)

1111(x), inscribed 1111(a)

In some cases, the (a) and (b) [or (a) and (x)] pairings can be recovered by inscribing a square in similar fashion. In other cases, the [(a) & (b)] or [(x) & (y)] sets are mirror-image pairs.

For the general case where the tiles are n-sided regular polygons with m entry points per side so the product of n and m is even, the number of distinct combinations can be computed as ${\displaystyle A(n,m)}$.[3]

${\displaystyle A(n,m)={\frac {1}{n}}\cdot \sum _{n=pq}\varphi (p)\cdot \alpha (p,mq)}$

${\displaystyle \varphi (p)}$ is Euler's totient function, where ${\displaystyle \varphi (p)=r\prod _{p\mid r}\left(1-{\frac {1}{p}}\right)}$

${\displaystyle \alpha (p,mq)}$ is a conditional function defined as:

${\displaystyle \alpha (p,mq)={\begin{cases}\sum _{0\leq 2j\leq mq}p^{j}{\dbinom {mq}{2j}}(2j-1)&{\text{if }}p{\text{ is even}}\\p^{\frac {mq}{2}}(mq-1)&{\text{if }}p{\text{ is odd}}\end{cases}}}$

For example, a 10-sided decagon with a single path per side (5 paths total) has 105 combinations.[3]

Number of unique tiles with n sides, each with m entry points[3]

		m Entry points per side
		0	1	2	3	4	5	6	7	8
n Number of sides	0 (circle)	1		1		2		5		18
	1 (monogon)	1		1		3		15		105
	2 (digon)	1	1	3	11	65	513	5363	68219	N/C
	3 (triangle)	1		7		3483		N/C		N/C
	4 (square)	1	2	35	2688	508277	N/C	N/C	N/C	N/C
	5 (pentagon)	1		193		N/C		N/C		N/C
	6 (hexagon)	1	5	1799	N/C	N/C	N/C	N/C	N/C	N/C
	7 (heptagon)	1		19311		N/C		N/C		N/C
	8 (octagon)	1	18	254143	N/C	N/C	N/C	N/C	N/C	N/C
	9 (nonagon)	1		N/C		N/C		N/C		N/C
	10 (decagon)	1	105	N/C	N/C	N/C	N/C	N/C	N/C	N/C

(Many of the values from the original source are N/C, meaning not computed)[3]

A game of Psyche-Paths requires one to six players. Players take turns laying down tiles. Each tile laid down next to tiles that have already been laid must continue the color of the path(s) on the existing adjacent edge(s). Six blank tiles are provided as "wild cards" and are considered to continue any adjacent path. One tile is drawn at the start of the game and placed in the middle of the playing field to "seed" the game.[7]

In "Beginner" Psyche-Paths, players draw one tile from a pool of face-down tiles when it is their turn. One point is awarded for each connected path. If the player is unable to play the tile or makes an illegal play, it is retained and the player is penalized a number of points equal to the number of paths running through the retained tile at the end of the game.[7]

In "Standard" Psyche-Paths, two to four players each draw a hand of six tiles at the start of the game. During each turn, players may play any or all of their tiles, replenishing to a hand of six tiles at the end of their turn from the pool of face-down tiles. When laying down tiles, points are awarded only when two or more ends of an existing path is connected. The player scores a number of points equal to the number of tiles containing the connected color path. Three points are awarded if the path crosses itself, and the score is doubled if the path is closed. No penalty is taken for leftover tiles.[7] "Classic" Psyche-Paths is similar to "Standard", but adds a rule that each move must result in a single path through all the tiles in that move.[7]

"Solitaire" Psyche-Paths does not have specific rules, but rather suggestions, such as building alternating rows of eight and nine pieces in length using legal moves.[7]

Square tiles are used in connection games such as the Black Path Game (with a single-colored path) and Trax (with two colored paths). Octagonal tiles with a single path color are used with interstitial spaces and a game board with pre-printed links for the game Octiles.[8][9]

The Serpentiles (2008) game was developed by Brett J. Gilbert[10] and published by ThinkFun in 2008.[11] The Serpentiles (2008) game includes 19 plastic tiles: 4 square (1×1) and 12 rectangular (2×1) tiles with one printed green or blue path on each, and 3 square (1×1) node tiles with coincident blue and green path termini. The 2008 game also includes 40 challenge cards which provide a list of tiles (including two of the nodes) to be arranged so that one contiguous path of each color is made.[12]