wickart@iuvax.UUCP (05/01/84)
I saw a simple version of this problem in a book called YOUR MAVE,
which I highly recommend to puzzle people. The general problem
(of my own definition) has been a source of much pleasure and
frustration over the years.
The problem is a two-player game. The players share a set of
racks, each of which can hold an infinite collection of balls.
The balls are numbered from 1 to aleph-null (omega). The first
player places ball 1 in one of the racks. Player 2 places ball 2,
player 1 places ball 3, player 2 places ball 4, and so on. A ball
may not be placed in a rack if the numbers on any two other balls
in that rack add up to the number on the one being placed. IE:
if balls 1 and 2 are both placed in rack 1, ball 3 must go in some other
rack, since 3=1+2. The player placing the last ball wins.
The number of racks is fixed at the beginning of the game.
The problems are:
1) Given the number of racks, find the best strategy for each player.
2) If the players cooperate, what is the largest number of balls that
can be racked?
3) in each of 1) and 2), express the solution as a function of the
number of racks.
The maximum number of balls for a 2-rack game is 8:
rack 1: 1 2 4 8
rack 2: 3 5 6 7
With 3 racks, there are several similar ways to rack 23 balls.
For 4 racks, I have a whole bunch of 52-ball solutions, but don't
have the resources to exhaust the possibilities without bothering
the powers that be. Can anyone else shed light on this problem?
T.F. Prune (aka Bill Wickart, ihnp4!inuxc!iuvax!wickart)wickart@iuvax.UUCP (05/01/84)
Correction. Title of book is "Your Move"--T.F.Prune