Another Sticky Problem

The game presented by a Java applet below was published in 2004 the Mathematics Magazine as problem #1687 (Sung Soo Kim). A solution by Li Zhou appeared in v. 78, n. 1 (February, 2005), p. 70.

A two-player game starts with two sticks, one of length N and one of length N+1, where n is a positive integer. Players alternate turns. A turn consists of breaking a stick into two sticks of positive integer lengths, or removing k sticks of length k for some positive integer k. The player who makes the last move wins. Which player can force a win, and how?

The applet allows for experimentation with different starting lengths. At the outset, you can force the computer to make the first move by pressing the "Make move" button. Sticks are represented by a row of small squares that might remind of a chocolate brick. To break a piece click on a square to the right of the desired break line.

Strategy

Copyright © 1996-2008 Alexander Bogomolny