Originally posted by elopawn
I'm thinking of a 6-digit number. The sum of the digits is 43. And only two of the following three statements about the number are true: (1) it's a square number. (2) it's a cube number, and (3) the number is under 500000."
No, I have a real answer. The number is 499,849, which is 707 squared and is less than 500,000.
The first pair of stipulations is easy to eliminate, because there are only three 6-digit numbers that are both squares and cubes, and none of them has a digit sum of 43.
So we know the number must be either a square or a cube, and less than 500,000. With one digit less than 5, the remaining digits would have to average out to between 7.6 and 8.4, which means there has to be a bunch of 8's and 9's in the number. I used Excel to check the 5-digit cubes with 6 digits (47^3 to 79^3), and did a quick scan. None of them had enough 8's or 9's, so I moved onto the squares. Luckily, the largest number producing a 6-digit square less than 500,000 is 707, which gives a square of 499,849 which has a digit sum of 43. Et voila!