If you had the following choice of salary methods, which would you choose and can you explain why this is your choice?
Option A: One billion dollars per year plus health coverage, a yearly expense account of $1 million, and a $100,000 car.
Option B: The amount of money obtained by putting $1 on one square of a chessboard, $2 on the next, $4 on the next, $8 on the next, and so on until all 64 squares are filled.
Students, working in small groups, typically attack this problem in the following way.
Step 1. Draw a chessboard and begin putting numbers in each square.
Step 2. As the numbers begin getting larger, students usually switch to a chart system like the following.
Square Total Money So Far
2 $1 + $2 = $3
3 $1 + $2 + $4 = $7
4 $1 + $2 + $4 + $8 = $15
Step 3. Once they see a pattern in the chart, they can generalize, that is, generate a formula. In this case, the formula is 2x - 1.
Finally, they can solve the problem by simply plugging in 64, for 64 squares, into the formula as x. The answer is $18,446,744,070,000,000,000. (This has been rounded off.)
A version of this article appeared in the January 01, 1996 edition of Teacher as Question and Solution