Sylver's coinage is an example of an unbounded finite (if slightly modified) combinatorial impartial game. Quoth wikipedia:

The two players take turns naming positive integers that are not the sum of nonnegative multiples of previously named integers. After 1 is named, all positive integers can be expressed in this way: 1 = 1, 2 = 1 + 1, 3 = 1 + 1 + 1, etc., ending the game. The player who named 1 loses.*

This can be made to have the normal play convention if we make $1$ an illegal move.

In Conway's ONAG, it is shown that Grundy's number can be generalized to ordinal numbers for unbounded impartial games.

**Is anything known about Grundy's ordinal for Sylver's Coinage and its various positions?**

_{*Wikipedia contributors. "Sylver coinage." Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, 21 Oct. 2014. Web. 27 Mar. 2015.}

arepositions known to have value $\omega$ and I believe there are some with value $\omega+n$ for various $n$), but don't hold me to either of those. $\endgroup$ – Steven Stadnicki Mar 27 '15 at 2:11