Good one! Getting the 16 from so little information is a neat trick, and after doing a little candidate work here and there, the 32 is pinned. Very elegant.

OK, here goes.

We have a free 2, and then the 16 must be a product, two of which are NOT 2. This means it can only be 4x1x4.

The 32 must be a sum, and since the other two regions it shares columns with can at most contain any particular number twice, the 32 contains at least one of each number. The other 6 cells must add to 17, and the four cells of the other regions in these columns add to 13. The 4 can have at most a sum of 6 (5-1), so the 2 has a sum of at least 7; this can only be 8 (5-3) so the 4 is 4x1. Then the last cell of row 2 is a 1 and the 4 can be filled in.

The 1 must be a difference, and can only be 3-2, so it can be filled in, the other cell in its row is a 5, and the 2 can be filled in.

The 8 cannot be a product 2x4 since these columns both have 4s, so it is the sum 5+3, and the rest can be filled in by elimination.

14235
42351
35142
23514
51423

You know, calling puzzles "easier" or "mediumer" is somewhat clumsy. If you want a "unique" difficulty ladder, try using the days of the week as the difficulty levels, with the guideline that you have a sliding scale of difficulty from Monday to Friday, with Saturday puzzles being more of a challenge, and Sunday puzzles being more of a challenge on different levels.

My use of any difficulty label came from how horrible the NYT difficulties are named with Easier, Medium, Harder. Changing Medium to Mediumer at least added consistency.

You are right that I don't need this label at all though, as I am scaling the challenge with the week, so I won't apply any label again.