This is the instuctor version of this article. For the student version, click here:
TipTheDie-Students.html
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?"
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.
!SPOILER ALERT! #1
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.)
(30)
If the total is 30 and anything but a 1 or 6 is showing, you play the 1 and win.
If the total is 30 and a 1 or 6 is showing, you lose because you are forced to go over 31.
(29)
If the total is 29 and anything but a 2 or 5 is showing, you play the 2 and win.
If the total is 29 and a 2 or 5 is showing, you might think you lose because you cannot play the 2. However, you can play the 1 and force your opponent over 31. Moral: Never give your opponent a 29 or s/he will win.
(28)
If the total is 28 and anything but a 3 or 4 is showing, you play the 3 and win.
If the total is 28 and a 3 or 4 is showing, you might think you lose because you cannot play the 3. However, more analysis is necessary. If you play the 5 or 6, you go over 31 and lose. If you play the 1 or 2 your opponent plays the 2 or 1 (resp.) and wins. So in any case you do lose.
(27)
If the total is 27 and anything but a 4 or 3 is showing, you play the 4 and win.
If the total is 27 and a 4 or 3 is showing, you might think you lose because you cannot play the 4. However, more analysis is necessary. If you play the 5 or 6, you go over 31 and lose. If you play the 1 your opponent has a total of 28, plays the 3 and wins as shown above. If you play the 2 your opponent has a total of 29, plays the 1 and wins as shown above. So in any case you do lose.
Continue in this manner, looking at each configuration of a current total and what number is showing on the top. For each configuration, try each number you can play and see what configuration that gives your opponent and whether your opponent has a winning strategy, in which case you would lose. Make a table of the winning numbers you can play for each configuration. You will eventually find that there are some totals for which there are no winning plays. These are the totals you want to give your opponent.
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.
!SPOILER ALERT! #2
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
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.