The Surprisingly Interesting Mathematics within Chutes and Ladders
By: Jake Mitchell
Tags: Board Games, Simulation, Mathematics, Matlab, Data Science

Results:

The average number of turns it takes to reach the 100th space in Chutes and Ladders is 38.93 turns, with a standard deviation of 24.81. Below is the histogram for number of turns per game.

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The best game of Chutes and Ladders involved just 7 dial spins. Starting from space 0, they spun a 1, which took them up a ladder to the 38th space. Then they spun a 5 and a 3, bringing them to the 46th space. Another 5 resulted in another ladder sending them up to space 67. A dial spin of 4 sent them up a third ladder to space 91. Finally, a dial spin of 6 and 3 put them right on the 100th space as shown below.

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The worst game of chutes and ladders went on for an incredible 312 turns. Even if every turn took you only 10 seconds to play, that would still take you 52 minutes to finish the game (assuming you were just playing by yourself). Below is the route that was taken during this horribly tedious game.

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As you can see, there were 9 different times when the pawn reached a space above 80, only for it to have to slide down back into the 20’s or 30's.

At this point, I started wondering if your starting spin had any affect on how many turns it would take you to win. Well it turns out, it does:

Spinning a 1 on starting turn: 35.482 average turns-to-win

Spinning a 2 on starting turn: 40.508 average turns-to-win

Spinning a 3 on starting turn: 39.713 average turns-to-win

Spinning a 4 on starting turn: 37.999 average turns-to-win

Spinning a 5 on starting turn: 39.977 average turns-to-win

Spinning a 6 on starting turn: 39.857 average turns-to-win

Spinning a 1 on your first turn can drastically decrease the expected number of turns to win. This is because it will send you straight to the 38th space, just as it did in the best Chutes and Ladders game mentioned previously. Spinning a 4 will give you better chances of winning, as that will send you up a ladder to space 14. Overall, a 2 is the worst value to start with because it doesn’t have a ladder connected to it, and it prevents you from reaching space 9 on the next turn (which does have a ladder).

Since your starting spin does have some affect on your expected turns-to-win, I wanted to know if your average spin value over the course of a game had any affect on your turns-to-win.

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It turns out that spinning lower or higher numbers on average does increase your chances of having a lower expected turns-to-win value. This doesn’t correlate 100% of the time (the best game had an average of 3.86 spin value), but it is interesting to see that simulations that had more 3’s and 4’s spun averaged longer games.

Just like I did in my Monopoly Simulation, I wanted to find out what spaces are most likely to be occupied after each turn. Those results are shown below:

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Spaces that are empty are the start of the ladder or a chute, so they do not ever end with a pawn on it. Spaces that are abnormally high are spaces that have the end of a ladder or a chute. As you can see, there is a higher curve on the spaces 20–40, and again once you get above 75. The 99th space is extremely high, because players get stuck on this space trying to spin a 1 to get exactly 100. This results in multiple turns ending without the player moving from the 99th spot.