### PUZZLE #3: THE REBATE GAME

This puzzle was devised by John Burrow as a computer simulation exercise for his students.

An antiques shop owner, in order to drum up more business, offers the following "rebate scheme" for anyone who makes a purchase:

(1) The purchase, costing a whole number of dollars, is first bought and paid for.

(2) With the real money exchange out of the way, the owner presents to the customer a bowl of plastic "coins," all marked in the same denominations as real money.

(3) The customer is told that she will be asked to select a coin randomly from the bowl, record the result, and replace the coin in the bowl. This procedure will be repeated until the sum of the results is enough to cover the cost of the purchased item(s). But first (and this is where the rebate part of the game comes in) she is told that she will be refunded the amount shown on either the first coin drawn or the last, and that she must declare beforehand which it will be.

The problem is to decide whether a "first coin" or a "last coin" strategy is most likely to maximize the rebate (or whether there is any difference at all in the two strategies).

(solution)