See Winning Ways for several disguises. R. On Tue, 8 Apr 2003, Roland Silver wrote:
I find the 3-dimensional 4x4x4 variant of TTT interesting. I don't know how to play it optimally, although it's clear that the first player shouldn't lose.
I published a paper on the automorphisms the board (permutations of the cells which preserve 4-in-a-row) some years ago (Amer. Math. Monthly; Vol. 74, No. 3 (March 1967), pp. 247-254). Aside from the obvious subgroup of 48 rotations and reflections, there are two other independent automorphisms. One scrambles the board by interchanging each of the 3 parallel pairs of inner planes, while the other inverts the board by interchanging each of the 3 pairs consisting of an inner plane and its parallel outer neighbor.
The whole group has 192 elements.
The peculiar thing about the 3x3 magic-square version of TTT is that if you memorize the magic square, you can usually beat people who you could never beat playing the geometric version. Eg you start off choosing 2, say; then when your opponent finds the effective counter, you switch to 4, then 6, then 8. He probably won't "see" the obvious isomorphic drawing strategy based on symmetry, which is obvious with the geometric version.
Of course you can play it by laying out 9 playing cards, face up (A-9 of hearts, say), with the 2 players alternately picking up cards until one gets three that add up to 15, or all the cards are picked up without a win.
Martin Gardner (I think) once suggested playing tic tac toe by instead having the two players choose integers from the set 1,2,3,4,5,6,7,8,9, the first player to obtain a sum of fifteen winning. The magic square
2 9 4 7 5 3 6 1 8
then proves useful.
My question: does anyone know a way to make tic-tac-toe, or something like it, into an interesting game? My sons, 5 and 7, are always wanting to play it and I'm suffering. I'm too strong at dots and boxes to play them with enjoyment.
Here's a problem for an idle moment:
How many essentially different "cat" positions (ie, fully played-out draws) are there in tic-tac-toe with best play, if we regard two as the same under a rotation or reflection of the board?
Thane Plambeck 650 321 4884 office 650 323 4928 fax http://www.qxmail.com/home.htm
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-- --Rollo <rollos@starband.net>
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