Count of day 6 misere-inequivalent impartial games
Chris Thompson
The context is counting impartial games inequivalent under misere play
(player unable to move wins).
On pp. 139-140 of [ONAG], John Conway writes "Counting only games in their
simplest form, Grundy and Smith showed that there was 1 game born by day 0,
2 by day 1, 3 by day 2, 5 by day 3, 22 by day 4, and 4171780 by day 5. We
extend their list one place by remarking that there are exactly
2^4171780 - 2^2095104 - 3.2^2094593 - 2^2094081 - 3.2^2091522 - 2^2088960
-3.2^2088448 - 2^2087937 - 2^2086912 - 2^2086657 - 2^2086401 - 2^2086145
-2^2085888 - 2^2079234 + 2^1960962 + 21
games in simplest forms born by day 6."
In a fit of ennui, I recently tried to verify that last number. However,
I have been unable to reproduce it as printed. What I get is:
2^4171780 - 2^2096640 - 2^2095104 - 2^2094593 - 2^2094080 - 3.2^2091522
- 2^2088960 - 2^2088705 - 2^2088448 - 2^2088193 - 2^2086912 - 2^2086657
- 2^2086401 - 2^2086145 - 2^2085888 - 2^2079234 + 2^1960962 + 21
Maybe I am overlooking some subtlety, or just persistently getting part of
the computation wrong. Is anyone else bored enough to do an independent
check?
If the result printed in [ONAG] is in error, then [WW] also falters when it
says (p. 397) of this number that it falls short of 2^4171780 by so little
that "the first 625140 of its 1255831 decimal digits are not affected", for
several hundred more digits are in fact affected...
[ONAG] "On Number and Games"
John H. Conway
Academic Press, 1976
ISBN 0-12-186350-6
[WW] "Winning Ways for your mathematical plays"
Elwyn R. Berlekamp & John H. Conway & Richard K. Guy
Academic Press, 1982
ISBN 0-12-091101-9
Chris Thompson
Email: ce...@cam.ac.uk