Reply #32: you were right the first time

[foo_bar]

The Marienbad nim is crooked in a way most other nims aren't, including this 1-2-3-4-5 example:

001 (1)
010 (2=2)
011 (3=2+1)
100 (4=4)
101 (5=4+1)
^^^ XOR
001

Since the nim-value* is nonzero, the first mover has two potential branches, instead of the deterministic flip to odd-parity in Marienbad's 1-3-5-7, per http://mathworld.wolfram.com/Marienbad.html

*The nim-value can also be found by writing the number of counters in each heap in binary, adding corresponding binary digits (mod 2), and interpreting the resulting binary string as a decimal number.