If each heap has its own last-number-taken counter, the Nim/Sprague/Grundy analysis below still applies [for sums]
The misere analysis may not go through for sums so simply (although it may; I haven't checked). Here's an example of a similar, probably simpler game that is easy to work out in normal and misere play for single heap positions (each one has grundy number 0,1, or 2 in both normal and misere play), but for which heavier weapons need to be shouldered to handle misere sums: On Pascal's triangle, place a bean. Two players take turns move it up and to the left, or up and to the right, until the boundary is reached (binomial coefficient 1). This can be solved for multiple beans in misere play but one has to drag out the genus theory of disjunctive sums (or something equivalent) to do it... Thane Plambeck 650 321 4884 office 650 323 4928 fax http://www.qxmail.com/ehome.htm ----- Original Message ----- From: "Scott Huddleston" <scotth@ichips.intel.com> To: "math-fun" <math-fun@mailman.xmission.com> Sent: Wednesday, July 30, 2003 10:06 PM Subject: Re: [[math-fun] Nim-like game]
If each heap has its own last-number-taken counter, the Nim/Sprague/Grundy analysis below still applies.