A deck of nine cards can be numbered, so that the sum of the numbers on a randomly chosen pair of cards totals to an integer from 2 to 12 with the same frequency as rolling two standard dice. What are the numbers on the nine cards?

If the wording doesn't specify, then you have to figure it out for yourself. Since (as pointed out before your post) 81 isn't divisible by 36, it must be without replacement.

My findings show with non-negative integers (with or without 0) it's impossible to do this:

Without 0, the only way to make 2 is 1 and 1. Then we need two ways to make 3 or 1-2, so the only way to do that is add another 2. Then, to make 4, we can only use 1-3 since adding another 2 would make 4 sums of 3. However, we can't get 3 sums of 4, only 2 or 4, so it's impossible without 0.

With 0, we can't have a 1 since that would make a sum of 1. So we must use 0-2 for 2. Then, we must use 0-3 for 3 and 0-4 for 4 since we can only have one 2 and no 1 to use. So, we have 0, 2, 3, 3, 4, 4, 4 currently, but only 2 sums of 5, so we must use 0-5 and use up the last two numbers allowed giving 0, 2, 3, 3, 4, 4, 5, 5. For 2, 3, 4, 5, 8, 9  the distributions are right, but we have 3 sums of 6 instead of 5, 8 sums of 7 instead of 6, 1 sum of 10 instead of 3, and no sums of 11 or 12. 

Posted by Gamer on 2004-08-09 18:28:07