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Roll of the die (Posted on 2005-04-21) Difficulty: 4 of 5
The object of the dice game is to be the first player to reach a score of at least 100 points.
Each player’s turn consists of repeatedly rolling a die.
After each roll, the player has two choices: roll again, or stop.
- If the player rolls 1, nothing is scored in that turn and it becomes the opponent’s turn.
- If the player rolls a number other than 1, the number is added to the player’s turn total and the player’s turn continues.
- If the player stops, the turn total (the sum of the rolls during the turn), is added to the player’s score, and it becomes the opponent’s turn.

What's your strategy?

See The Solution Submitted by Hugo    
Rating: 4.1111 (9 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Endgame: Q = 98 or 99 | Comment 8 of 23 |
Dustin is completely right that one should be more conservative when ahead and more agressive when behind.

Unfortunately, Dustin, I think the function which says when to stop rolling is not as simple as just being a function of the difference between the two scores.  I believe that it is of the form f(P,Q), where P is your score at the start of your turn and Q is the opponent's score.

This sure is complicated to get exactly right, though.

I have exhaustively examined the case where Q = 98 or 99.  In this case, if you yield the die, your opponent has a 5/6 probability of winning on the very next roll.  I have worked out all cases, and even if you have started with 0 (98 or 99 points behind), you have a chance of just about 3/1000 of winning, as long as it is your roll.  You have to keep rolling (hitting), because there is no point where stopping and banking your points improves your probability of winning. 

Here is a sample calculation, from my very elaborate spreadsheet.  Let's say that your opponent has 98 points, you had zero at the start of the roll, and you have rolled an incredible 80 points.  Should you keep rolling?
If you stop, you will lose immediately 5/6 of the time, and the other 1/6 of the time (if the opponent rolls a 1) you will have a .3372 chance of winning.  This works out to a .0562 chance of winning, if you stop.
If you roll, you have probability
1/6 of rolling a 6 -- your chances have improved to .4015
1/6 of rolling a 5 -- your chances have improved to .3781
1/6 of rolling a 4 -- your chances have improved to .3562
1/6 of rolling a 3 -- your chances have improved to .3358
1/6 of rolling a 2 -- your chances have improved to .3164
1/6 of rolling a 1 -- your chances have dropped   to .0005
Overall, if you roll again, your chance of winning is .2808.
Rolling again (and risking your 80 points) is 5.5 times better than stopping.

I'm not sure how far to take this.  I would like to examine one case where I can calculate exactly when to stop.  Maybe Q = 97 will do it, but I doubt it.  I might need to work my way down to Q = 80, or even lower.
  Posted by Steve Herman on 2005-04-23 04:33:21
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