A magic square contains positive consecutive integers starting from 1 such that the rows, columns, and diagonals all add to the same value.
A heterosquare contains positive consecutive integers starting from 1 such that the rows, columns, and diagonals all add to different values.
If the sums resulting from a heterosquare form a consecutive sequence, the heterosquare is also called an antimagic square. (In the example above, if the sums of 9 and 24 were 16 and 17 instead, it would be a antimagic square.)
Prove that there are no 3 by 3 antimagic squares.
Unlike with Open Problem #1, this has been proven by computer brute force. The task here is to prove the result by something other than brute force.
For the second Open Problem I wanted to emphasize the fact that mathematics doesn't just emphasize unknown pieces of knowledge; it also improves existing knowledge. A brute force proof doesn't give much insight into the nature of a mathematical structure, so a more elegant proof of the lack of 3 by 3 antimagic squares would be quite helpful for mathematicians working on the subject.
A related (but likely more difficult) open problem is: does an \( n \) by \( n \) antimagic square exist for all \( n > 3 ?\) The answer is thought to be "yes" but nobody knows for sure.
Since heterosquares are a superset of antimagic squares, one approach to this would be finding a systematic way of categorizing all heterosquares, and then looking for the special properties heterosquares that are not antimagic have.
It's also possible to approach from the brute force computing end (even thought it's been done before) and work on ways of gradually improving the algorithm for identifying antimagic squares.