# Solution 1 May 29, 2007

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### Problem

Does there exist two unfair 6-sided dice labeled with numbers 1..6 each such that probability of their sum being j is a number in for each ?

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### Solution

This is impossible. Let the probabilities for the sides of the first die be and similarily let be those for the second die.

The probability for the dice showing a sum of 2 is . This implies . Similarily, is the probability of a sum of 12, and we have . For the probability of a sum of 7 we have

This is a contradiction; multiplying both sides by and bringing all terms to one side gives

which is false. This completes the proof.