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Custom Dice (Posted on 2006-11-16) Difficulty: 2 of 5
You are given blank six-sided dice, with equal weight on each side (fair dice). You're allowed to write on each side of each die a number between 0 and 6. Numbers may repeat themselves on a single die, so for example, one die can have five 4s and one 0.

1) Can you create two such dice which, if they are rolled and their results added, will give equal odds for every number from 1 to 12?

2) Can you create three such dice which, if they are rolled and their results added, will give equal odds for every number from 1 to 18?

(NOTE: Don't get smart and make all the dice all 0s. The probability of getting 0 should be 0)

No Solution Yet Submitted by TamTam    
Rating: 4.0000 (1 votes)

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Solution re: Extension to question 2 Comment 5 of 5 |
(In reply to Extension to question 2 by Joel)

Using a technique from Renumbered Dice, I find 6 sets of integer dice:

{9,9,9,0,0,0},{2,2,1,1,0,0},{7,7,4,4,1,1}
{9,9,9,0,0,0},{3,3,2,2,1,1},{6,6,3,3,0,0}
{10,10,10,1,1,1},{2,2,1,1,0,0},{6,6,3,3,0,0}
{1,1,1,0,0,0},{4,4,2,2,0,0},{13,13,7,7,1,1}
{1,1,1,0,0,0},{5,5,3,3,1,1},{12,12,6,6,0,0}
{2,2,2,1,1,1},{4,4,2,2,0,0},{12,12,6,6,0,0}

These come from the nonnegative factorizations of the generating function (the coefficient 12 comes from 216 possible rolls/18 possible values = 12 ways for each possibility)
12*(x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^10+x^11+x^12+x^13+x^14+x^15+x^16+x^17+x^18)

  Posted by Brian Smith on 2017-02-27 12:56:16
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