I have an unfair die whose sides are known to have the following probabilities:{.02,.04,.08,.12,.24,.5}
Assign the numbers {1,2,3,4,5,6} to these sides so the expected roll is the same as for a fair die.

Can you solve this without brute force?

I managed to find 4 solutions by hand. To start with, I multiplied both sides (the sum of the probabilities and 3.5) by 50 so I was working with integers, which gave:

a + 2b + 4c + 6d + 12e + 25f = 175

I then worked out the minimum and maximum possibilities with f being 1..6. This eliminated f=1 and f=6 and gave definite solutions for f=2 (134562) and f=5 (643215).

Playing guess and check with f=3 and f=4 I found 246153 and 526134, but had no idea if they were the only ones or not. Evidently (see Charlie's comment) not!

Posted by snark on 2006-05-03 22:29:34