Serpentile Games

Serpentiles allow the creation of richly patterned or random-seeming mosaics from relatively small tilesets. Here we'll take a look at a couple games that make use of a simple set and their colored variant.

Setup

For this activity, you will want a set of square serpentiles. I've created some for printing out in this document. You can use plain paper, but they'll be nicer with cardstock. For DIY game pieces like this, I've often printed the whole sheet on cardstock, laminated it, and then cut out the pieces with an X-Acto knife. Though the edges are exposed, the tops and bottoms being laminated gives the pieces a good amount of rigidity and longevity.

One activity will make use of the black and white tiles and the game boards. I have provided the game boards in \(5 \times 5\), \(6 \times 5\), \(6 \times 6\), \(7 \times 5\), \(7 \times 6\), and \(7 \times 7\), which should give a good sampling. The other activity will make use of the colored tiles and will not require a game board.

Brick

This game is sometimes called the Black Path Game and a variant of it was popularized by Martin Gardner, citing William L. Black as the originator. (In the version Gardner presented, there is a target square to win in, which we'll forego in favor of winning by not losing.)

In Brick, the first player must play one of the two tiles in any rotation somewhere along the edge of the game board. This starts a path leading from the edge to the interior. Players take turns placing tiles that elongate this path, always playing at the square adjacent to its end. When a tile is placed that runs the path into any edge of the board, the player who placed that tile loses. The goal is to not be that losing player.

Players should alternate being first and second player as they work on their strategies. The first player has the advantage on some of the included boards, while the second player has the advantage on the others.

Example

Below is an example game on a \(5 \times 5\) board. Moves read left-to-right, top-to-bottom, and it is organized so that the left column shows moves by the first player and the right column shows moves by the second player:

Since the first player was forced to steer the path into the wall in their last move, the second player won this game.

Loop Trax

Trax is a game conceived of by David Smith. I'm presenting Loop Trax because it has a win condition that's easier to spot, though the game may last longer.

In Loop Trax, the first player plays as red and the second player as blue. The game begins with the red player placing a tile on the table. Players then alternate placing tiles adjacent to an existing tile so their paths line up with colors matching (red-to-red or blue-to-blue). If a player places a tile in such a way that it leaves an empty space that can only be filled by one specific tile, that player must also place that tile before the turn is over. (This can set off a chain reaction of tile placements in a single turn.) The first loop to form wins the game for its color's player. If placing a tile creates loops of both colors, the player who placed that tile wins the tie.

As usual, players can trade back and forth the roles of first/red player and second/blue player as they devise their strategies.

Example

Below is an example game. Moves read left-to-right, top-to-bottom, and it is organized so that each row shows the chosen move and subsequent forced moves, if any, made by a player in a single turn:

The first loop to form was blue, so the second player won this game.

Notes

• In Brick, the first player has the advantage on an \(m \times n\) board if at least one of \(m\) and \(n\) is even while the second player has the advantage if they are both odd. A nice strategy due to Berlekamp involves tiling the board with dominoes:

  • If one of \(m\) or \(n\) is even, tile the board with dominoes any way you like. If the first player always uses their turn to steer the path to the center of a domino, then the second player has no choice but to play in the other square of that domino, finishing the pair of squares. This means that there are no half-dominoes on the second player's turn except the one they must complete, so no matter what the second player does, their turn will steer into an untouched domino, giving the first player the ability to again path to its center. Since the border of the board must fall along the border of a domino and not its center, the first player will never extend the path to a border.

    Here is an example of back-and-forth play, where the first player steers to the center of a domino and the second player completes that domino:

  • If \(m\) and \(n\) are both odd and the first player moves in a square of the right parity, the remaining \(mn-1\) squares can be tiled with dominoes. The second player then uses the above strategy of steering the path to the center of a domino.

• Due to the requirement that colors match, the kinds of arrangements one can get in Trax are different than those in Brick. In particular, a path cannot cross itself in Brick, because the path segments in crossings are different colors. From the way we chose to have red cross over blue, this means that every game will look like a layer of non-intersecting red curves and loops lying over a layer of non-intersection blue curves and loops.

• In Loop Trax, forced moves make up a large part of the strategy. In our example game above, you can see that if there is a \(2 \times 1\) domino that has same-color paths leaving through both increments of a long side into an empty \(2 \times 1\) domino, then a player can connect these together using a move and the forced move it demands. Below are a couple examples of a configuration, and the move and forced move a player can make. Here's one connecting red paths:

  Here's one connecting blue paths. (The red paths could also have been connected on the left, instead.):

  The way the blue player won in our example game relied on setting up two forced moves. This relied on a \(3 \times 1\) rectangle with one side featuring blue, red, blue feeding into an empty \(3 \times 1\) rectangle. Here's another example of using this to join two blue paths with a red path between them:

  Since these can be moves that close loops, players can use these sorts of tricks to close their own loops but must also defend against the other player using these to their advantage.

• Trax and its variants are far more sophisticated games than Brick, so strategies that humans follow tend to be far more heuristic. This page has a few strategy pointers for beginners.

Extra Challenges

Originally, serpentiles were made from hexagons and there are a variety of games that have been played using them. These include:

• Kaliko/Psyche-Paths

• Tantrix/The Mind Game

When running this sort of activity with a math circle, I will usually do something like play Brick or Loop Trax and then ask, "How could this game work with hexagonal tiles?" For example, we've seen there are essentially only two square tiles that can be made: opposites edges paired or adjacent edges paired. Any tile is a rotation of one of these. How many different hexagonal tiles are there? How does this change if we use different colors for the paths? Students can design their own pieces and then play a game with them. For hexagonal Brick, they'll need to create a game board, which is another interesting design challenge.

If we allow two path entry/exit points per side, we can get other interesting sorts of tile sets. For example, triangular tiles become viable. The square tiles of this type are used in the game Tsuro, a multiplayer game about keeping your path disjoint from other players' paths and not falling off the edge of the board.

This post was sponsored by the Julia Robinson Mathematics Festival