Die Game

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Grade Vs Difficulty:
  EasyModerateChallengingPerplexing
1-2
3-4
5-6
7-8
9-10
11-12
13-14

This game came from a Renaissance fair via Phil Yasskin.

The die game is played as follows: A goal number is chosen (usually 31). The first player places the die down with some number pointing up to start the game. The number on the top is the first sum.

After the first move, players take turns making moves. A move consists of rolling the die 90 degrees in any direction. The number that ends on the top is added to the sum to get a new sum. Play continues as long as the new sum is not more than the goal. The last player to make a move wins.

For example if the goal is 31
the first player can put the die with a 6 on top. The second player can roll
the die to any number other than 1 or 6 (so might try 4 bringing the total to 10).
Play could continue as follows: Player 1 puts 6 up sum is 16; Player 2 puts 2 up, the sum is 18; Player 1 puts 6 up, the sum is 24; if player 2 now puts a 2 up the sum will be 26 and Player will be able to force a win. (Think about this.)

This is a fun game to play. It is good practice at addition for young children. There are a number of interesting further questions that can be asked:

What are optimal moves. Figure out who will win (Player 1 or Player 2) if the game is played optimally for each goal. The optimal moves are a periodic function of the goal.
What is the period?

The attached "lesson plan" is a spoiler. It just gives the answers to the above questions.