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. (The related question of if \( n \) by \( n \) antimagic squares exist for all \( n > 3 \) is entirely open.)
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.
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.
Information about what antimagic squares look like in the larger cases would help understand the conditions for one to be made.
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.
Some last comments (for now) on Open Problem #1:
Thanks for everyone who contributed! Our "final" wiki page giving results is here. I will be sharing this with the mathematician who originally wrote the problem (Erich Friedman) so there may be another update in the future!
Also, while the group effort will likely be focused on the current Open Problem, you are still welcome to work on the old one. Please label any posts that relate to an old Open Problem quite clearly so people don't get mixed up.