quaternary games

Note added 27 June 2005
Quaternary Bounties (PDF, 2 pages)

Stalking the Woolly Mammoths

The Quaternary is subdivided into two epochs, the Pleistocene (1,600,000 to 10,000 years ago) and the Holocene (10,000 years ago to the present). The Pleistocene Epoch thus comprises almost all of Quaternary time...
—Encyclopedia Britannica

But we're not interested in that quaternary.

Quaternary Games

Quaternary games are octal games whose octal codes involve the (base four) code digits 0, 1, 2, and 3 only. They generalize subtraction games, which have code digits 0 and 3 only.

Subtraction games always reduce to nim heaps in misère play (see Winning Ways), but a quaternary game sometimes has non-nim heap positions. A complete theory for them is not yet known.

Winning Ways contains a table showing that the only two-digit quaternary that does not contain only nim-positions is the game .31, called Stalking:

Quaternary periods

We've extended this analysis to three-digit quaternaries, finding that all these games have only misère nim-heaps as values, except for the seven Woolly Mammoths

.121, .123, .131, .201, .211, .312. and .331.

Two of the woolly mammoths, .201 and .211, are first cousins of Stalking (a heap of size n in Stalking plays exactly as a heap of size n+1 in .201 and .211). We've been able to solve the others, too, by pretending that their position heaps are equivalent to appropriate sums of adders and (in the case of .123, only) appropriate restive and restless games with additional simplification rules. So, unlike two digit quaternaries, some three digit quaternaries contain non-tame positions.

See the links at the left hand side of this page for more information on specific quaternary periods. The (PDF format) notes called How to Lose at .123 give more background and explanation of some possible methods of solving such games.