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#### Posers and Puzzles

1. 24 Mar '08 06:44
Before I start I have to do some definitions first:
(1) In an ordinary six-sided dice two opposite sides always adds to seven.
(2) A quarter turn is when you turn the dice into the nearest side.
(3) There are only four different quarter turns.

Okay, here is the game of 'First to 31'

The object of the game is to reach 31 exactly, and not over.
If you reach exactly 31 then you win and your opponend lose.
If you can force your opponent over 31 then your opponent lose and you win.

First: You throw the dice and note the outcome.
Then: Your opponent make a quarter turn and add this outcome to the earlier sum.
Then: You make a quarter turn and add... and so on...
Until: Any of you get exactly to 31, and win. Or get over 31, and lose.

Are the rules well defined?

The questions here are - Is there any strategy how to win this game?
Who will win if this strategy is employed from the beginning?, the one throwing the dice first or his opponent?
Or is this a fair game from the beginning?
2. 24 Mar '08 20:30
can you stop if you're close to 31?
because if you don't get 31 and you are first you lose!
first look conclusion: it seems easyer to make your opponent lose
3. 24 Mar '08 21:18
Originally posted by alexdino
can you stop if you're close to 31?
because if you don't get 31 and you are first you lose!
first look conclusion: it seems easyer to make your opponent lose
No, the game is stopped only once one reach 31 or go beyond.

Say your opponent gets to 30 with a 6, then you can't reach 31 with a 1, because 1 is the opposite side of 6. You have to chose 2, 3, 4, or 5 and you're lost.
4. 25 Mar '08 01:49
Analysis by Excel indicates that so long as the initial roll isn't a 4, the first to turn the dice can force a win.

Trying to find a concise way to state the strategy though..
5. 26 Mar '08 15:40
Originally posted by geepamoogle
Analysis by Excel indicates that so long as the initial roll isn't a 4, the first to turn the dice can force a win.

Trying to find a concise way to state the strategy though..
There could be a strategy, you mean, but it is hard to verbalize?
6. 26 Mar '08 20:37
great puzzle, fabian!
I modified my endgame tablebase algorithm ( hey, this is a chess players site! ) to find a solution
and came to the same result as geepamoogle, but of course much more sophisticated than using excel ;-)
I printed out all possible positions ( = position+last dice-side),
but cannot find a pattern.
There are some positions completly won ( sum= 13 or sum =22)
and much more completly lost ( sum in 29, 25,...). The 3rd group
have always a 4+2 pattern.

I cannot evolve a winning strategy, but I am working on!

Btw you are doing a great job here in the forums, I don't understand, why
rhp does not sponsor guys like you the membership
7. 27 Mar '08 19:02
Here's a breakdown of the good picks for any situation in this problem. (Essentially I solved this by means of endbase tables as well, Excel was just a tool to keep information straight.)

SUM = 4,13,22: No Winning Move
SUM = 1,10,19: Turn to 3 or 4
SUM = 2,28: Turn to 3
SUM = 3,12,21: Turn to 1 or 5
SUM = 5,9,14,18,23,27: Turn to 4
SUM = 6,15: Turn to 2, 3, or 4
SUM = 7,16,25: Turn to 3 or 6
SUM = 8,17,26: Turn to 5
SUM = 11,20: Turn to 2 or 3
SUM = 24: Turn to 2, 3, 4, or 6
SUM = 29: Turn to 1 or 2
SUM = 30: Turn to 1
8. 28 Mar '08 06:32 / 2 edits
No programs, just good old-fashioned chartmaking.

When you (or your opponent) are faced with a sum on the left with a number from the top showing on the dice, a W indicates a win and an L indicates a loss.

1,6 2,5 3,4
31: L L L
30: L W W
29: W W W
28: W W L
27: W W L
26: W L W
25: W W W
24: W W W
23: W W L
22: L L L
21: W W W
20: W W W
19: W W L
18: W W L
17: W L W
16: W W W
15: W W W
14: W W L
13: L L L
12: W W W
11: W W W
10: W W L
9: W W L
8: W L W
7: W W W
6: W W W
5: W W L
4: L L L
3: W W W
2: W W W
1: W W L

As for filling out this chart, the highest few numbers were obvious. As you get lower, you have to go through each number one by one. Suppose I only had the numbers 25 to 31 filled in (which is not very hard to do). For 24, I would check the 1,6 column of 25 (since that's what my opponent would have should I choose to switch the dice to 1) and see that there's a W for my opponent, so 1 doesn't work. I would then check the 2,5 column of 26, which is an L, so 2 works. Then the 3,4 column of 27 shows an L, so 3 also works. If necessary, I would have then checked the 3,4 column of 28, the 2,5 column of 29, and the 1,6 column of 30. Whenever two numbers that are not opposites work, a W should be placed in all three columns, as it was with 24. If one number works, or two opposite numbers work, then a W should be placed in two columns, but not the one containing those numbers (since, if the dice showed one of those two numbers, I would not be able to choose either of them). If no numbers work, an L should be place in all three columns. As was pointed out earlier, the only time this occurs in the first roll is at 4. So for any other initial roll, the roller wins, but for an initial roll of 4, the rollee wins.