Playground construction

            Keeping up with the spirit of tweaking the conditions of the game and observing how the length of the game is effected, this week I also wanted to mess around with the placement of chutes and ladders. Normally we think of chutes as harmful and ladders as helpful, but this, perhaps, mustn't always be the case. For instance, if a short chute puts a player a few squares away from a very long ladder that the player had already passed, that chute may actually be advantageous. Can we find a case on the Chutes and Ladders board for which this is true?

            To find out how a chute or ladder impacts the length of the game, we can remove it (which alters the transition matrix) and see how the expected number of moves changes. I started with what seemed to be the most likely candidates for these "good" chutes and "bad" ladders. The chute from square 47 to square 26 caught my eye since it puts the player in a good position to land on the behemoth ladder on square 28; however, removing this chute still decreased the expected number of moves (from 39.2 to 36.8). My second choice was the chute from square 16 to 6, but removing this also decreased the expected number of moves (from 39.2 to 38.0). After checking the remaining cases, no such chute or ladder could be found.


            So, then, how about we create one ourselves? Let's see what happens if we put a chute from square 29 to square 27. This makes it a bit more likely for the player to land on the ladder at square 28 without setting them back too far, and it decreases the expected number of moves from 39.2 to 38.0. What happens if we instead make the chute go from square 29 to square 26? The expected number of moves drops again to 37.9. Even though this set the player back further, the increased chance that this gives him or her of landing on the big ladder makes up for it. In fact, this chute has the best effect if it goes from square 29 to 23, where the number of moves set back is equal to the size of the spinner (expected number of moves is 37.2).

           I want to generalize this result a bit more and perhaps establish under what conditions a chute or ladder will be "good" or "bad", but for now, that's all folks.