The Exact Chaos Spreadsheet Game

John D. Cook points out the following interesting fact:

Pick a number x between 0 and 1. Then repeatedly replace x with 4x(1-x). For almost all starting values of x, the result exhibits chaos. Two people could play this game with starting values very close together, and eventually their sequences will diverge.

Cook's on to something here. This does have the makings of a simple game that ties in to math and computing concepts like functions, iterations, spreadsheets and chaos. Here are the rules:

The first player picks a to number from 1 to 99 with the only condition being that the number can't have a five in it (with a little trial and error you should be able to figure out why).

The second player can either steal the first player's number or come up with one of his or her own. If the number is stolen, the first player has to come up with another number.

Both numbers are divided by one hundred and put into the first row of a spreadsheet. Under the first type


=4*A1*(1-A1)


Under the second

=4*B1*(1-B1)



Now copy the formulas going down twenty rows. The copy function should produce an iterative function, with =4*A1*(1-A1) followed by =4*A2*(1-A2) and so on.

Have the students make a line graph of the two series. The winner is the one who ends up with the largest number.