You spin me right round...

    This week I was finally able to indulge my childish tendencies by breaking the rules of Chutes and Ladders.

    In a normal game of Chutes and Ladders, the spinner has a "size" of six: there are six possible moves. This week I decided to figure out what spinner size would produce the lowest expected number of moves to complete the game. While it might seem that increasing the spinner size would allow us to advance along the board faster, and hence, reduce the expected number of moves, increasing the size also makes landing on the final square more difficult. The ideal value is therefore in the middle of two extremes.

    Thankfully, two weeks ago I prepared a program to find the expected number of moves for a standard game with a spinner of size six, and finding this value for other spinner sizes only requires a small tweak in how the transition matrix is constructed. The results were as follows:

Spinner size -- Expected number of moves

1 -- Infinite (the game cannot be completed with a spinner of size one)
2 -- 60.7
3 -- 65.9
4 -- 54.5
5 -- 45.6
6 -- 39.2
7 -- 34.7
8 -- 31.8
9 -- 30.3
10 -- 28.8
11 -- 27.4
12 -- 27.0
13 -- 26.2
14 -- 26.0
15 -- 25.8
16 -- 25.9
17 -- 26.0
...

    The results continue to shoot up after this, so the ideal spinner size is 15.

    As you can see, the ladders and chutes cause some irregularities in the pattern (most noticeably when the spinner has a size of 3). This begs the question: what would be the ideal spinner size for a modified game of Chutes and Ladders with no chutes or ladders? A similar analysis shows that the ideal size is then 13. Thanks to the simplified nature of this version, however, this value can be found without using any complicated Markov chain analysis. The expected number of moves for a game of spinner size n can be approximated by:

         The first fraction is an approximation for the expected number of moves required to be within n moves of the finish. "n" is also the expected number of moves required to reach the final square when you are within n moves of that square (the probability is always 1/n, so the expected number of moves is n).

    Plugging in various values of n, again, reveals that the ideal spinner size is 13.

    Let us take a moment to breathe in and enjoy this brief respite from the pain of Markov Chains.