[Cogito]

There are many strict orderings that can be used, but we just need one.

Since each number is in a physical draw in a physical room, and no two draws can occupy the exact same space, there is a strict ordering of the drawers given by each drawers distance from a fixed point in the room (say a specific corner agreed on beforehand) in each of the three orthogonal dimensions sideways (x), backwards (y), and up (z).

We say a drawer A is before a drawer B if A.x < B.x; or, if A.x = B.x then if A.y < B.y; or, if A.x = B.x and A.y = B.y then if A.z < B.z

If A.x = B.x and A.y = B.y and A.z = B.z then A = B.

This is a strict ordering that will number the drawers from 1-100, and can be determined beforehand.

Now the prisoners take that order, and shuffle it randomly, without telling the warden what they are doing.

[Retric]

As math that works fine, in practice it's a world of problems. A.x = B.x is a question of finite precision close enough and all prisoners will not agree if A.x = B.X or A > B or A < B. Adding precision tools does not help as you get into things like thermal expansion and the weight of prisoners deforming the floor etc.

So, while that works as a math problem, it may fail as an actual solution.