IBM Ponder This #337 · May 2026

This riddle was proposed by Lorenzo Gianferrari Pini and Radu-Alexandru Todor - thanks!

Given a square binary matrix of order $N$, $A\in M({\mathbb{Z}}_2{)}_{N\times N}$ such there is exactly one "1" value in each row and column of $A$, we can find the lowest $m>0$ such that $A^m=I$, the identity matrix.

Denote by $g(N)$ the maximum such $m$ when going over all the matrices in $M({\mathbb{Z}}_2{)}_{N\times N}$ satisfying the above condition.

For example, $g(10)=30$ and $g(50)=180180$.

Your goal: Find $g({10}^6)\text{mod}({10}^9+7)$.

A bonus "*" will be given for finding $g({10}^8)\text{mod}({10}^9+7)$.

Solution

Best opened after a real attempt

To be added.