I’ve got a game for you. Two, really: one for snake-haters, and one for snake-lovers.
The first comes from master game designer Sid Sackson. Encountering it in his writing was like finding a new creature in the underbrush, an unknown reptile, with its own strange form of locomotion.
The game-taxonomist in me delighted. Sackson called it “Hold That Line,” but I call it…
The Game of Snake
Players: Two, although under conditions of extreme boredom or lack of paper, another one or two could join.
What You Need:
- A pen
- A four-by-four array of dots (or larger, if you like)
- A healthy fear of snakes (that’s ophidiophobia, for my Ancient Roman readers)
The Goal: Force your opponent to complete the snake.
How to Play:
- One player begins by connecting any two dots via a vertical, horizontal, or diagonal line. Here are three possible opening moves:
- Now, players take turns growing the snake from either end, using horizontal, vertical, or diagonal lines, like so.
- There are some restrictions. The snake (a) must never cross itself, (b) must never revisit a used dot, and (c) must grow only vertically, horizontally, or at a 45o angle.
- Eventually, the snake can grow no further. At this moment, it springs to life, and discharges its hateful venom into the most recent hand to touch it. Or, more prosaically: Whoever completes the snake is the loser.
It’s a breezy yet strategic game. If you’re nimble, you may even discover a guaranteed winning strategy. Unfortunately, this rather undermines the fun.
(No spoilers here, but I’ll offer two hints: (1) Try it on a 3-by-3 array; and (2) Try the variant where the person who makes the last move is the winner. The solution to this version be adapted, without much trouble, to the traditional game. For more analysis, see Jim Henle’s discussion, and the pertinent MathOverflow thread.)
In any case, once you have a handle on the flow of Sackson’s original, you’re ready for my preferred variant: Snakes!
(Aliases include “Snake Breeder” and “Snakes in the Coordinate Plane.”)
The Game of Snakes
Players: Two, though if you want to add a third, just use a larger board and another pen.
What You Need:
- Two pens (different colors)
- A five-by-five array of dots (or larger; any size works)
- A logic-defying love of snakes (parseltongue fluency encouraged)
The Goal: Draw as many snakes as possible.
How to Play:
- Play proceeds much as in Snake. (First, you begin a snake. Then, on each move, you connect a free end of the snake to an unused dot, via a vertical, horizontal, or 45o line, all without crossing lines or reusing dots.) But there’s a key difference: in this game, each player is growing their own personal snake.
- Moreover, in this version, you want to finish your snake as fast as possible. That’s because, when it can no longer grow, you get to begin a new one.
- Eventually, a player will be ready to begin a new snake, but have no space to do so. In that case, the other player simply finishes their snake, and the game ends.
- The winner is whoever created more snakes. If it’s a tie, then look at each player’s snakes, and count up the number of dots. Whoever used fewer dots is the winner.
In this game, both players made two snakes, but blue used fewer dots (11 vs. 13), and thus is the victor.
Once you get the hang of it, the choices you face are subtle and satisfying. Nothing is sweeter than stealing a dot that your opponent was relying on, thus forcing them to go careening off into the open board.
(Nothing, that is, except completing a late-game snake in just a single move. That’s triumph itself.)
The multi-snake variant also spawns a solitaire version—or, really, a collection of puzzles. With no opponent, just creating snakes on your own, how many snakes can you pack into a board of a given size?
For example, in the 3-by-3 board, the best I can achieve is two snakes:
On the 4-by-4 board, meanwhile, I can manage four snakes:
On a 2-by-n board, you can fit n/2 snakes if n is even, and (n+1)/2 snakes if n is odd.
At this point, I cede the floor. Open questions:
- What other “snake numbers” can you figure out? Can you prove that they are optimal? Is there a formula for the 3-by-n board, or the 4-by-n?
- What strategic gambits can you devise for the two-player version? Is there a learnable winning strategy? Are there useful heuristics?
- Who wins in the two-player version, if both players move optimally? Does it depend on the board size?