# Snakes, ladders…everyone’s a winner!

(From the Archives of The Science Bit)

23 December 2013

I’ve been playing a lot of Snakes and Ladders these past few days. I know it’s a game of pure chance, but I feel I’ve been getting quite good at it. Wikipedia has a nice piece on S&L, including a section on its mathematical aspects. Go take a look — I find this type of graphic particularly helpful.

Basically, it shows that you’re always increasingly likely to win the more you play. You can hardly win after your first move, or your second. But give it a few more and that chances-of-winning-line just shoots on up. It’s statistics, so it is.

Random environments have their beneficiaries. In fact, because the game is one of pure chance, all players are equally likely to win (although the spoils of victory are seldom lucrative). Over the course of 100 games between two players, you should see very close to a 50:50 split in victories. It almost looks as though when one person wins, their opponent’s chance of subsequent victory duly increases.

Imagine what would happen if, by fluke, Player A were to win the first ten games in a row. In order for the 50:50 split to occur, Player B would need to win a corresponding accumulation of ten “catch-up” games from the remaining 90, as well as the baseline 40 other victories, thereby requiring a deviation from the presumed 50% chance level (which would’ve predicted 45 victories from 90 games).

It’s almost as though the randomness fairy has to step in to make those 90 games, well, less random. But of course this doesn’t happen. When you toss a coin and get Heads, the odds of the very next coin toss showing Tails is still 50%. It’s the same with Snakes and Ladders. Being kick-ass at S&L for a few games doesn’t mean anything. It’s all just a fluke. And yet, over 100 games, you’ll see two opponents slugging it out to a tie most of the time. Weird.

Snakes and Ladders represents an absorbing Markov chain (I particularly agree with the “absorbing” bit). This means that the odds of progress are unconnected with game history.

I also like the fact that progress in Snakes and Ladders is of the back-and-over variety, such that you move left-to-right on row X before moving up to row Y where you move right-to-left. And so on. Orthographically, this is referred to as a boustrophedon trajectory, which is Greek for “turning your ox (as when ploughing)”. As I say, I like this.

My 3-year-old opponent loves it when I share such wisdom.

S&L FTW!

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