# The game Officers

In the book "Winning Ways" by Berlekamp, Conway, Guy (aka the bible of combinatorial game theory) there is a short section about the game Officers in Chapter 4. It has also the symbolical name $\cdot {\bf 6}$. Its Grundy function is given by

$G(n) := mex(\{G(a) \oplus G(n-1-a)\} : 1 \leq a \leq n\}$,

where $\oplus$ is the nim-sum. This is the OEIS sequence A046695, where you also find some values. It is noted that this sequence has "a strong inclination towards a period of $26$", but that "a complete analysis is still to be found".

Question. What is known nowadays about this game? Specifically, is it known that it is periodic?

According to the paper "Periods in Taking and Splitting Games" by Ian Caines, Carrie Gates, Richard K. Guy, and Richard J. Nowakowski, this was still open in the year 1999. Does anything has changed so far?

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What are the rules? –  TonyK May 18 '12 at 15:31

## 2 Answers

Guy and Nowakowski indicate that it is still open in a paper dated 2008.

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Thanks! This is an interesting paper. –  Martin Brandenburg May 19 '12 at 16:51

Achim Flammenkamp's list of octal games (most recently dated as of November 2012), seems to report the exploration of $2^{33}$ positions without a proven solution.

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