If ... you must play in the [only] zero pile, winning the game, does that eliminate all loops, yielding a game that has Grundy values?

Many plays reduce the zero population. The exceptions are plays into an Edge Sequence: a non-empty sequence of ones at an edge of the playing board. Such a play moves the bounding zero closer to the edge, and shrinks the Edge Sequence. If there is only one zero, that move expands or creates the other Edge Sequence: the game can loop. If there are more zeroes, the move doesn't affect the other Edge Sequence, and reduces the sum of those two sizes. The number of such moves is bounded, and then one must reduce the zero population. To do a slowest game, always move to the leftmost box, (N^2+N)/2 times. ... Ah, I didn't solve Jim's game before the deadline. But I did solve the vastly simpler game played on a circle instead of a line. >:-> -- Don Reble djr@nk.ca