Pig

Pig is a two-player dice game of mathematical interest. It was first described by John Scarne in his 1945 book Scarne on Dice, but it is a rather simple game and, like other simple games such as Nim, its roots are probably older.

The rules of Pig are pretty straightforward. You need a single die. Players roll the die and add the number of points on the die. You don't need to alternate turns. You can keep rolling as long as you want, or you can pass the turn to your opponent at any time. However, if you roll a 1, you lose your entire score for the turn and have to pass the turn to your opponent. The first player to reach 100 points or more wins.

A variant of Pig for two dice goes as follows. The players roll two dice at a time, adding the total of the two dice to their score. As above, you don't need to take turns. However, if a 1 turns up on one die, you lose your score for the turn and have to pass the turn to your opponent. If a 1 turns up on both dice, you lose your entire score for the game so far and have to pass the turn to your opponent. As above, the first player to reach 100 points or more wins.

Several other versions of Pig exist. You may find it interesting to invent your own version and figure out how to play it well.

Obviously, knowing addition is important when playing this game, but to do well a basic knowledge of probability (e.g. what your expectation for the next roll is, or what the probability of something bad happening) is necessary to help to understand how to win. Finding an optimal strategy is more intricate than that though. How do you decide whether to keep rolling or to pass the turn to your opponent? Does this depend on how many points your opponent already has, or what score you have, or what score you have for the turn so far? In 2004, Todd W. Neller and Clifton Presser, of Gettysburg College, analyzed the one-die version of Pig and discovered a strategy for optimal play. The strategy is very complex and depends on the considerations mentioned above.