General bimagic squares of order 5

Find numerical examples and, if possible, construct a parametrized family of general bimagic squares of order 5.

Background

The introduction states that general bimagic squares of orders 3 and 4 are impossible, whereas the existence of a general bimagic square of order 5 is not known. The concluding remarks explicitly identify finding numerical examples, and possibly a parametrized family, of order 5 as an open problem.

References

Further, as noted in the introduction, bimagic squares of orders less than 5 do not exist, but there remains the open problem of finding numerical examples and possibly a parametrized family of general bimagic squares of order 5.

— Construction of parametrized families of general pseudo-bimagic and bimagic squares  (2610.00915 - Choudhry, 1 Oct 2026) in Section 5, Concluding remarks