There are no choices to make in the game of Chutes and Ladders (also known as snakes and ladders or snakes and arrows). Therefore, it is an easy game to simulate. These analyses simplify things further by considering only a single player's journey to square 100, using the Milton Bradley layout of the board. All the code used to produce the figures is in main.py.
Figures 1 and 3 show histograms of two different simulations of 20,000 games. The number of turns to win ranges from 7 to over 300, with the peak around 30. It is clearly not a Gaussian distribution.
Figure 2 examines the statistics of simulation sets more closely. The mean, median, mode, and standard deviation for the number of turns from 5000 simulations was recorded and repeated 200 times. The box plots of Figure 2 show how these statistics differ across the 200 sets of simulations.
Note that the mean, median, and mode do not overlap at all, as the long tail of long games skews the results. The long games pull the mean up to around 39, while the median number of turns is around 32-33. But the most common number of turns is lower, varying from 17 to 30 across the different simulation sets.
Future work includes modeling this distribution as a Poisson distribution, possibly with modifications due to the fact that the minimum number of turns is 7, not 0.
The dreaded space 87 sends the poor player the whole way back to square 24. It seems that the more times one lands on this chute, the longer it takes to complete the game. Figure 4 shows that this intuition is correct. In this sample of 500,000 simulations, the longest chute was traversed as many as 11 times, and these games could last hundreds of turns.
Games with many trips down the long chute are rare, however. Figure 5 shows that 0 is by far the most common number of times to land on square 87, while 9, 10, and 11 occurred less than 100 times each in the 500,000 simulations.
Future work may include calculating a model for the relationship between the number of times down the longest slide and the length of game.
In short, no. Some chutes and ladders are traversed more frequently than others. See Figure 6. The chutes and ladders are labeled by their starting square.
