The open problem and its partial forms
The 3×3 magic square of squares
Does there exist a nondegenerate order-3 magic square whose nine distinct integer entries are all perfect squares? Over the ordinary integers, this problem remains open.
The general form
Every 3×3 magic square is uniquely determined by three coordinates E, x, and y. Its center is E and its magic sum is 3E:
The full 9/9 problem requires all nine linear forms to be distinct nonnegative integer squares. The laboratory exposes E, x, and y directly and checks the resulting square.
Why the atlas starts with 4/9 and 5/9
Instead of assuming an unproved 9/9 solution, we fix a set of cells whose values are guaranteed to be squares. Other cells may also happen to be square-valued: 4/9 and 5/9 describe a guaranteed mask, not necessarily the exact number of square entries.
Rotations and reflections form the group D4. Modulo this action there are 23 four-cell masks and 23 complementary five-cell masks.
What the proof contains
- a complete decomposition into D4 orbits and a proof that no further orbits exist;
- the original equations in E, x, y, and the square roots;
- elimination of E, x, and y to one quadric for 4/9 and two independent quadrics for 5/9;
- explicit parametrizations, applicability assumptions, and the widest proved coverage domains currently available.