|
|
[Sitemap]
|
|
Why are Series Musical? Why are Series Musical?ASKS BLANCHE DESCARTESMOST mathematicians know the theory of the game of
Nim, described in books on mathematical recreations. But few seems to
be aware of Dr. P. M. Grundy's remarkable generalisation, published in
Eureka, 2, 6-8 (1939).
Consider a game
It follows that if Now imagine the players engaging in a "simultaneous display" of
k games
For no player can gain any advantage by moving so as to increase any G(Ps), as the opponent can restore the status quo. And if only decreases in G(Ps) are considered, the game is identical with Nim, thus proving assertion (i). Therefore G(P) = g if and only if the combined position (P, P') is safe, where G(P') = g. From that (ii) follows fairly readily. It follows that we can analyse any such combined game completely,
provided that we can find the G(Ps) for the component
positions. Nim is an example; a heap Hx of
x counters constitutes a component position, since each
player in turn alters one heap only, and G(Hx) = x.
Many variants of Nim are similarly analysed. Less trivial is Grundy's
game, in which any one heap is divided into two unequal (non-empty)
parts. Thus heaps of 1, 2, are terminal, with G = 0, a heap of 3 can
only be divided into 2 + 1, which is terminal, so
G(H3) = 1. Generally G(Hx) in
Grundy's game is the least integer
continuing with 3, 2, 1, 3, 2, 4, 3, 0, 4, 3, 0, 4, 3, 0, 4, 1, 2, 3, 1, 2, 4, 1, 2, 4, 1, 2, ... This curious "somewhat periodic series" seems to be trying to have period 3, but with jumps continually occurring. Mr. Richard K. Guy confirmed this up to x = 300. He suggested that it might be played on a piano, taking 0 to be middle C, 1 = D, 2 = E, etc. The inner meaning then became evident:
Guy also worked with rows Rx of x counters, in which certain sets of consecutive counters could be extracted (thus possibly leaving two shorter rows, one each side of the extracted set). In his ".6" game, any one counter can be removed, except an R1 (= a single counter standing on its own). The G(Rx) series (x = 1, 2, ...) is a waltz (N.B. some notes span two bars):
But at this point the tune completely broke down. I asked Guy if he could think of any reason for that: he said, "Yes, an error I made in the calculation." After correction the waltz proceeds:
This tries to be periodic with period 26, but jumps keep appearing.
Many other such games give tuneful, somewhat periodic series, for no
evident reason. Guy discovered two curious exceptions: his
".4," remove 1 counter not at the end of a row, has exact
period 34 for Eureka, 16. Reproduced from Eureka 27 pages 29-31. |
|