Inventory012:  Another variant of nim (proposed by Stephen C. Locke, 11/14/01).

This problem is based on my misreading of Problem 714 of CMJ.
Two players P_0 and P_1, play Nim with one extra feature. Before P_i moves, P_{1-i} specifies exactly one Nim-pile, H_j, and one integer, m. P_i is then forbidden from taking exactly m from the pile H_j (although he would be allowed to take m from another pile the same size as H_j, if one exists). Determine a winning strategy for this game.
The version in the journal allows (as I now read it) the blocker to list an integer for each pile, rather than for just one. I cannot estimate the difficulty of this question. It might be no harder than the regular Nim strategy, or it might be completely very sporadic. (I assume that the one in [1] is barely more difficult than regular nim.)
It might make a good problem for a high school student to spend a semester on in preparation for a science fair.

Discussion.

Bibliography. [1] Problem 714, College Math. Journal, 32 (2001) 382.