Tip the Die, a Two-Person Game

Philip B. Yasskin

Department of Mathematics
Texas A&M University


This is the instuctor version of this article. For the student version, click here: TipTheDie-Students.html


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


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.


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:

Second Sample Game:


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?"

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


Postpone reading this section if you want to develop your own strategy.

Hopefully, the students will realize they should work backwards:
(Like you do to develop a strategy for the Subtraction Game or NIM.) I have written a Maple program to do this analysis of the strategy.
Do not look at these files, if you want to complete your own strategy either by hand or by computer. But you may want to keep reading after these links for some generic comments.
The Maple program including the output is linked here: TipTheDie.mw
The code and the output are shown in the .pdf file linked here TipTheDie.mw.pdf

Since opposite sides of a die always total to 7, the same results occur 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. In the output table, the first column shows the current total while the top row shows the what number is showing on the top of the die. Each entry in the table shows the possibe winning plays. If it says NULL, then there are no winning plays. (In the top left corner of the table there is an extraneous NULL which is a space holder and should be ignored.)

If you examine the output you will see that there are 3 current totals that have no winning play. 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 output, you will see that there is a periodicity in the table. 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.


Postpone reading this section if you do not want to know the periodicity or hear a discussion of the output table.

If you examine the output you will see that there is a periodicity of 9. Starting with a goal of 31 and subtracting 9, 18 and 27, there are losing rows when the current total is 22, 13 or 4. The table is pretty complex to remember, but if you can leave your opponent with 4, 13 or 22, you have a better chance of winning without any more strategy. Also notice that statring with the 4th row from the top (line 27), each row is identical to the row 9 below it. The top 3 rows are different because of the endgame requirement of not going over 31.

So what is the strategy if you leave your opponent with 4, 13 or 22? If your opponent plays 3, 4, 5 or 6, you play 6, 5, 4 or 3 (resp.) to get to the next period of 9. (This is like the Subtraction Game.) If your opponent plays a 1 or 2, you play a 4 and put your opponent in a losing position. (I don't really know why a 4 works except that the table says it does. Comments are welcome.) With this much strategy you can always win once you give your opponent a 4, 13 or 22.

A good host will always let their opponent choose whether to go first or second. If you get to go first, what do you do?
If the initial roll is a 1, 5 or 6, then the first player should play a 4.
If the initial roll is a 2, then the first player should play a 3.
If the initial roll is a 3, then the first player should play a 5.
If the initial roll is a 4, then you don't want to go first.
If you don't get to go first, or the initial roll is a 4, you have to finesse your way to 13 or 22. If you opponent does not know the strategy, s/he will usually do something to allow you to get on track.

Note: Some of the entries in the bottom 4 rows of the table do not make any sense. If the current total was the initial roll or the result of 1 tip, then only certain numbers are possible on the top. The entries in the other columns are extraneous.

Game Variations

Have fun.

Last Updated: April 18, 2015, PBY
Copyright © 2015 Philip B. Yasskin