Susan gave her nephew a number of pennies, as well as a mathematical challenge: to figure out how many ways there were of dividing the pennies into three piles. The pennies are indistinguishable, so the identity of the pennies doesn't matter, nor does the order of the piles. For example, if there had been nine pennies, the piles could have been arranged in any of seven ways: 1+1+7, 1+2+6, 1+3+5, 1+4+4, 2+2+5, 2+3+4, 3+3+3.
There were actually more pennies than this, and in fact, the number of ways was a four-digit number.
However, the nephew misunderstood the instructions. He thought that no two of the piles could be equal, and so came up with a smaller number. For example, if the number of pennies were nine, as above, only three of the arrangements into piles consisted of unique sizes: 1+2+6, 1+3+5, 2+3+4, and the nephew would have reported that, incorrectly.
As mentioned the actual number of ways was a four-digit number. The number reported by the nephew was also a four-digit number, and as a result of his misunderstanding, the only difference between his reported number and Susan's expected answer was that the middle two digits were reversed.
How many pennies did Susan give to her nephew?
After messing around with this for far too long today, I've arrived at the following tentative answer:
The number of pennies that Susan gave to her nephew is 182. These pennies can be arranged in three piles in 2760 ways. If we require the three piles to be distinct from each other, there are only 2670 ways to arrange them.
2760 = 2+3+5+6+8+9+11+12+...+86+87+89+90
2670 = 2+3+5+6+8+9+11+12+...+86+87+89
I hope this is right! Thanks for the puzzle Charlie; it's a good one.