A bogosolver is a valid approach. I mean it'll take forever, but it's valid.

You can solve it via brute force if you cut off branches that don't satisfy the rules:

https://github.com/tonyedgecombe/Sudoku/blob/master/Sudoku/P...

It's not 1ms but it's not that slow either.

I plead the halting problem.

There are a limited number of permutations. Assuming it's not choosing them randomly, it'll finish eventually... If the hardware doesn't die first.