You will have an opportunity to play the following game against the computer. We start with a pile of rocks. Two players take turns removing rocks from the pile. Each time, a player may remove 1, 2, or 3 rocks. The player who removes the last rock wins.

In this version of the game, the computer always goes first. (It's only fair, since you are far smarter than the computer.) The rocks remaining in the pile are indicated by the solid black disks. The rocks taken by the computer are indicated by red circles, and the rocks taken by you are indicated by the blue circles.

To take 1, 2, or 3 rocks from the pile, just click on the corresponding number. The computer will then automatically make its move. Try it out!

The computer's strategy is based on congruences. Can you discover what it is, and then beat the computer? You can vary the number of rocks at the beginning of a game by changing the "Starting Number" value and then clicking "Restart".

## Research Question 2

(a) For which starting values can you win, and for which starting values does the computer always win? What is the computer's winning strategy?(b) Suppose that at each turn, a player may remove 1, 2, 3, 4, 5, or 6 rocks. What strategy should the computer use, and what strategy should you use to beat the computer?

Section 4.1 | Section 4.2 | Section 4.3 | Section 4.4 | Section 4.5

Copyright © 2001 by W. H. Freeman and Company