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Renumbered Dice (Posted on 2004-07-20) Difficulty: 3 of 5
Two standard dice are renumbered so that each die is different, but together the pair still gives the same probabilities for rolling sums of 2 to 12 as a standard pair.

If each die is numbered with integers all greater than 0, then what are the numbers on the renumbered dice?

  Submitted by Brian Smith    
Rating: 3.1667 (6 votes)
Solution: (Hide)
The alternate dice are numbered [1 2 2 3 3 4] and [1 3 4 5 6 8]

The probability distrbution for standard dice can be generated by powers of the polynomial d(x)=(x+x^2+x^3+x^4+x^5+x^6). The coefficient of x^k gives the frequency out of 6^n.

For two dice: (x+x^2+x^3+x^4+x^5+x^6)^2 = x^2+2x^3+3x^4+4x^5+5x^6+6x^7+5x^8+4x^9+3x^10+2x^11+x^12

An alternate factorization of the polynomial is (x+x^3+x^4+x^5+x^6+x^8)*(x+2x^2+2x^3+x^4)

The first factor represents a die with faces [1 3 4 5 6 8]. The other factor represents a die with faces [1 2 2 3 3 4].

Gamer found a site with a similar solution here.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
These dice are commercially available :-)Percy2005-11-16 17:05:18
Solutiongot it with computervectorboy2004-07-26 16:24:24
Solving wayGamer2004-07-20 15:33:53
Solutionummm...Thalamus2004-07-20 12:06:25
Problem EditBrian Smith2004-07-20 11:39:52
Solutionadd/subtractAdy TZIDON2004-07-20 08:46:28
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