Apropos of nuclear pennies (which I'll call the "1+x^2 <--> x" game, for reasons that will be clear to anyone who read the "A Neighborhood of Infinity" web-page on the topic), here are some similar games: 1) The "2x --> 1+x^2" game. This is also called chip-firing, and a lot is known about it, although a lot is still mysterious: see the article "Disks, Balls, and Walls: Analysis of a Combinatorial Game" by Anderson, Lovasz, Shor, Spencer, Tardos, and Winograd (American Mathematical Monthly 96 (1989), 481-493; available on JSTOR and probably elsewhere as well). (I don't know whether anyone has explored the 1+x^2 --> 2x game. If you start with 2n+1 pennies in a row and start consolidating them via this rule, what are the terminal configurations that can result?) 2) The "x <-> 2" game. This is a sort of variant of chip-firing that can be used to construct the real numbers via their binary representations; see the article "The real numbers as a wreath product" by Faltin, Metropolis, Ross, and Rota; Advances in Math. 16 (1975), 278-304). It's somewhat amazing to me that the construction of the reals via radix expansion isn't better known. Everyone thinks it must be incredibly ugly, but with the right approach it's no more complicated than cuts or Cauchy sequences. 3) The "3x <--> x^2+2" game. This is related to what I call the "unary-binary system" of representing positive integers, via strings of digits like 10.2 that means the same thing whether you interpret them as extended unary or extended binary (1x^1 + 0x^0 + 2x^(-1) is three regardless of whether x is 1 or 2). Jim Propp