Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
What are the rules? – TonyK May 18 '12 at 15:31
up vote 3 down vote accepted

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

share|cite|improve this answer
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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.