ABSTRACT

You’re strolling past a casino and you notice an eye-grabbing sign:

The Multiplication Game, Carpe Diem!

Y ou’re curious, so you go into the casino to find out whatthe rules of the game are. The game is played at a table 31

between a player and a casino employee. The employee presses a button to generate a slip of paper printed with a four-digit integer on its reverse side. The player may not see this integer until he chooses one for himself. The player must choose a positive integer having as many digits as the player wishes. The player’s integer is then multiplied together with the four-digit casino integer. The player wins if the product of the two integers begins with a 4, 5, 6, 7, 8 or 9; otherwise, the casino wins. If the player wins, he gets 2.45 dollars for every dollar staked. This sounds tempting, seems too good to be true. Before deciding whether or not to play this game, you go off to calculate for yourself the possible products of two four-digit integers, from 1000 to 9999, in case you should play using a four-digit integer. Your computer program alerts you to the fact that 43.0% of the products begin with 4, 5, 6, 7, 8 or 9. This would mean that, in the long run, you will win on average 0.43×2.45−1 = 0.0535 dollar for every dollar staked, giving you on average a winning margin of slightly more than 5% for any fourdigit integer chosen. It certainly looks as though the casino has blundered, but hey, that is not your problem. You hightail it to the nearest casino to claim your winnings. After playing the game a great number of times, however, you are baffled to find yourself on the losing end of things. How could this happen? Your calculations were correct, and indicated a comfortable winning margin, but there you are, losing. Unfortunately, this is, in fact, the expected outcome. No matter what strategy the player uses, in the long run the casino wins at least 60.1% of the games. The trick is that the casino uses randomization in generating its four-digit integer. Randomization is a technique that has many applications in mathematics and computer science. For every interaction, i.e., every time the game is played, the random-number generator on the casino’s computer picks a randomly chosen number u between 0 and 1. It uses this number to calculate a number a = 10u and prints out the largest four-digit integer below 103× a on the casino’s slip of paper, generated at the start of the game. In this way, the casino guarantees itself winning odds of at least log10(4)−10−3 = 0.60106, no matter what strategy the player applies. This truly surprising outcome was discovered by American mathematician Kent Morrison.