The infinite z-list has only one ending in a 6. A move consists of subtracting a nonzero square from the current value, producing a new current value for the other player. You may not subtract more than the current value. The player who reaches 0 wins. In fact the only legal move from 2 is to subtract 1 leaving 1 for the other player.
Every natural number is either a winning value or a z-value, but never both.
Zugzwang is a chess term indicating a position wherein the obligation to move is a disadvantage. 1. 0 is the first z-value. IF it is greater than some z-value on the list by a nonzero square. Perhaps this has something to do with the fact that both 5 and 10 can be expressed as the sum of 2 different squares. A cursory statistical analysis of the distribution of z-values with respect to other modulus bases seems to corroborate this.