>>So, do you know how to prove whether a shuffle is winnable?

You only need to determine whether the permutation is odd or even. Lloyd's puzzle can be solved either by rows or by columns (i.e. numbers in the solution go 1-5 in first row, 6-10 in second row etc, or they do so in each column). But to know which way can it be solved, you first need to know whether it's an odd or even permutation of the solution - which is the reason Lloyd wasn't able to patent it. He couldn't prove it would work each time.