Tip the Die, P. Yasskin, 2015

Tip the Die, a Two-Person Game

Philip B. Yasskin

Department of Mathematics
Texas A&M University
yasskin@math.tamu.edu

2015

This is the student version of this article.

Introduction

Tip the Die is a two player game similar in style to the Subtraction Game or NIM but with more (or just different) strategy.

History

I don't know much about the history of this game. My son has worked at the local Renaissance Festival since high school. Several years ago, he came home and said one of the staff was playing a game after hours and no one could beat him. He asked me if I could analyze the game and give him a winning strategy. When I looked at the game I found it has a very interesting structure and complex strategy. While it has a similar structure to the Subtraction Game or NIM, it is significantly more complex. He said that the game was called the Game of 31, but since the number 31 is not significant to the strategy, I have come to call it Tip the Die because you tip a die to one of the adjacent sides without turning it over or rolling it.

Rules

Start by rolling a six-sided die. The number on top is the current total which is shared by both players and will grow as the game proceeds. The players alternate turns. On a player's turn, the player tips the die to one of the four sides other than the top or bottom, adds the new top to the current total and announces the new current total. The goal is to be the player who makes the current total become 31. If a player makes the current total go over 31, s/he loses. If the previous current total is under 31, the player must move. So the winner is the person who makes the current total become 31 or forces their opponent to go over 31.

First Sample Game:

The initial roll is a 5.
The first player can play anything other than a 5 (top) or 2 (bottom).
The first player plays a 3, the total becomes 8.
The second player can play anything other than a 3 (top) or 4 (bottom).
The second player plays a 2, the total becomes 10.
The first player plays a 6, the total becomes 16.
The second player plays a 4, the total becomes 20.
The first player plays a 1, the total becomes 21.
The second player plays a 5, the total becomes 26.
The first player cannot play the 5 to make 31 because it is on the top.
The first player plays a 3, the total becomes 29.
The second player plays a 2, the total becomes 31.
The second player wins.

Second Sample Game:

The initial roll is a 5.
The first player plays a 3, the total becomes 8.
The second player plays a 2, the total becomes 10.
The first player plays a 6, the total becomes 16.
The second player plays a 2, the total becomes 18.
The first player plays a 1, the total becomes 19.
The second player plays a 5, the total becomes 24.
The first player plays a 3, the total becomes 27.
The second player plays a 2, the total becomes 29.
The first player plays a 1, the total becomes 30.
The second player plays a 2, the total becomes 32.
The first player wins.

Strategy

After you play the game a couple of times you will see there is a lot of strategy to the game. The real question is "What is the strategy for developing a strategy?" You will have a strategy when you know what play or plays to make for each current total and number showing on the top.

In analyzing the game, there are a couple of simple facts you should know:

The opposite sides of a die always total to 7. So the same plays are possible if a 1 or 6 is on top, if a 2 or 5 is on top or if a 3 or 4 is on top.
Think about how you analyzed the Subtraction Game or NIM.
This strategy is different from the Subtraction Game, because in the Subtraction Game you can subtract the same numbers on every turn, whereas in Tip the Die the allowed numbers change; you cannot add the numbers on the top or bottom.

After you find a strategy, there two facts you should notice:

If you examine the strategy you will see that there are 3 current totals that have no winning play (as in the Subtraction Game). So if you can leave your opponent with one of those numbers, you have a significant advantage even if you know nothing else about the strategy.
Further examining the strategy, you will see that there is a periodicity in the strategy based on the current total. Since we are dealing with dice, one might expect that the periodicity might be 6 or 7. It is not and I do not really understand why it is what it is. If anyone can explain the periodicity (and not just that this is what you find when you analyze it) I would love to hear.

Game Variations

A slight varient is to have a different number other than 31 as the goal. The strategy is exactly the same except that everything is shifted up or down appropriately. For example, if the goal is 41 then all decisions in the strategy are shifted up by 10.
Finally, you might want to analyze what happens with other size dice. With a tetrahedral die, there is no number on top, so you have to add in the number on the bottom. With an octahedral, dodecahedral or icosahedral die, there are (at least) two different types of games, depending on whether you are allowed to tip only to put an adjacent face on the bottom or any face except the top.

Have fun.
Phil