Chutes and Ladders without Chutes or Ladders
The board game Chutes and Ladders dates back to ancient India, but was popularized in the United States by the Milton Bradley company in the middle of the twentieth century. The basic setup of the game is simple: There are squares numbered from one to one hundred, and players use a spinner to obtain an integer generated uniformly at random between one and six. They then move forward this number of spaces. The winner is the first player to make it through the complete board, but there are two caveats. First, the player must land exactly on the last square, and if they overshoot, then they lose their turn. Second, landing on certain squares causes the player to move either up the board via a "ladder" or down the board via a "chute".
Some mathematicians have attempted to model this game and to answer different questions about it. As far as we can tell, Daykin, Jeacocke, and Neal  were the first authors to model the game using the Markov process, a technique that several others have used in the past. More recently, Cheteyan, Hengveld, and Jones  used a Markov model to explore how varying the size of the spinner would affect the expected length of the game. In particular, they noticed that long spinners have higher expected values that allow one to move to the end of the board faster, but at the same time, such spinners make it harder to land exactly on the final square and finish the game. Their work found where the trade off occurred between these two competing ideas. For the traditional board, they find that the optimal spinner size is 15. [excerpt]
Glass, Darren D., Stephen K. Lucas, and Jonathan S. Needleman. "Chutes and Ladders without Chutes or Ladders." In The Mathematics of Various Entertaining Subjects: Volume 3: The Magic of Mathematics, edited by Jennifer Beineke and Jason Rosenhouse, 119–138. Princeton: Princeton University Press, 2019.
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