# The Game of Snakes

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:

1. One player begins by connecting any two dots via a vertical, horizontal, or diagonal line. Here are three possible opening moves:
1. Now, players take turns growing the snake from either end, using horizontal, vertical, or diagonal lines, like so.

1. 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.

1. 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:

1. 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.

1. 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.

1. 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.

1. 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.)

## Solitaire Snakes

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?

## 16 thoughts on “The Game of Snakes”

1. Aydin Hodala says:

Another brilliant blog post as ever! However, the n/2 rule for 2 x n does not hold. In the case of 2 x 4 you can fit 3 snakes in, if you constuct two of the three point triangles (shown in the 3 x 3 board) so that they cover a 2 x 3 rectangle, you can fit in a two-point snake at the bottom to give three snakes in total. In fact this “packing triangles” strategy allows for n/2 + 1 snakes for even numbers greater than 2. This method doesn’t work for odd numbers though because of the extra row, so an even number of rows seem to give the same number of snakes as the odd n.

2. Aydin, what a delight to hear from you! How’s your new decade going?

On the 2-by-n board, I actually think the three-point approach runs into trouble, because the first snake is not actually complete with three dots. For example, if I start in the upper-left dot, connect it to the dot below, and then connect this to the dot that’s above and to the right, my snake is not yet complete – one end can still be extended.

Or perhaps you were picturing something else? I need to figure out how to make WordPress allow image attachments in comments…

1. Aydin Hodala says:

My decade is starting off with university exams, so it could be going a little better to say the least. Luckily your blog posts always offer an interesting and welcome reprieve. I hope your year (and decade) have started a little more chreerfully!
Oh I see, I was growing my snake from one end only rather like the video game Snake. In which case, my solution doesn’t work and it is n/2, with your square spirals packing best.

1. Will Gaines says:

That is quite a key condition I think. Like Aydin I was working on the assumption that you had to grow from the same end, which makes the starting point for your snake a vital choice (especially in the Solitaire versions). If you can grow at both ends this condition is obviously less significant.
Can partial credit be earned for 2 snakes on a 3×3 board where one point goes unused?

2. Scott Nicholson Kurland says:

Ah! I was thinking of it as sequential, since snakes generally don’t have a reverse gear. So 241 would trap the snake even though the 2 end was free.

3. Marcello Bertocchi says:

hello Ben, this is Marcello from XVgames, a board game publisher based in northen Italy. This year i’m publishing a book about pen&paper games and I would love to put your game in it, if you allow it. If you are interested please contact me
thanks and have a nice day

4. Allie says:

If I understood the rules correctly, I was able to get three snakes on the 3×3 board. If you number the dots 1-9 then you have the following sequence of moves
Red: 1-5, Blue: 2-6, R5-7, B6-3 done, R7-4 done, B8-9 done.

1. Tim says:

Or in fewer moves: Red 1-7, Blue 2-8, Red 3-9. With this approach, an m×n board (with m ≤ n) can always have n snakes.

1. Will Gaines says:

As I understand it with the solo version you have to complete each snake before you can commence a new one? Rather than treating it as a 2-player game with alternating moves. Otherwise in the 2 x n boards you could make every snake length 2 and stack them to get n snakes every time?

1. Tim says:

Oh, I see. That does make more sense!

5. Francis Campena says:

very nice and interesting idea. It might be a good problem for undergrad math students or even a master student in math wanting to work on games on graphs.

6. Endles amusement until the snake eats itself. Oh wait, the rules don’t allow that.