XOR Solution

OK, anybody who cares has certainly solved the number puzzle from last time. Lets work through the solution now.

For convenience, I'll state the five facts here

1. N has 2 digits XOR N is even
2. N contains the digit "7” XOR N is prime
3. N is the product of two consecutive odd integers XOR N is one more than a perfect square
4. N is divisible by 11 XOR N is one more than a perfect cube
5. N is a perfect square XOR N has 3 digits

Notationally, I'm going to use each statement with a letter 'a' or 'b' to refer to the first part or the second part of the statement. Specifically, when I say "statement 4a" I mean "N is divisible by 11", and "statement 2b" means "N is prime" and so on.
Alright, most people latch right onto 1 and 5 right away, to figure out how many digits the number has, and that is a sensible thing to do. Also note that 5a and 3b are exclusive, since a number cannot be a perfect square and one more than a perfect square (except for 1, I suppose, but it is easy to verify that 1 is not a solution to the puzzle). Looking at 3a, being the product of two consecutive integers means N=m(m+2) for m odd. That is N=m²+2m=(m+1)²-1 for m odd. So 3a says that N is exactly one less than a perfect even square. Thus, by 3, N is next to a perfect square, and so 5a is false.

This gives us that 5b is true and so 1a is false and 1b is true. But since N is even, and 3a says it is odd, 3b must be true. So we have N is even, has 3 digits, and is one more than a perfect square. Letting K=N-1, K=m² for m odd. Consider 4b now, if that is true then K is a perfect cube and so being a perfect cube and a perfect square means it is a perfect sixth power. We know 2⁶=64, 3⁶=729, and 4⁶=4096, so the only consistent thing would be if K=729. This would be N=730, which satisfies 2a and not 2b, while satisfying 4b and not 4a. This is the answer.

I recall some more elegant method of eliminating the false statements in 2 and 4, but I cannot remember it now. As usual, feel free to comment with any alternate methods you have.