Marbles and Change
Old marbles challenge question
You have 10 marbles, half black and half white, to put in two bags any way you want. Your opponent (without looking) will select a bag at random and choose a marble at random from that bag. You win if he picks a white marble. What are your chances of winning? What if another enemy looks and switches one pair of marbles between bags before your opponent makes his selection?
For the best chance place one white marble in one bag and the rest of the marbles (4 white and 5 black) in the other. If your opponent chooses the first bag, he is 100 percent sure to choose the white marble; if he chooses the second bag, he still has a 4/9 or about 44 percent chance of a white marble. Therefore the overall chance of a white marble is about 72 percent (the average of 100 percent and 44 percent).
If a foe traded the white marble for one of the black marbles, the chance would drop to about 28 percent (the average of 0 percent and 56 percent). But you could anticipate the foe by putting three white marbles alone in the first bag. Then even after your foe switches one of them with a black one, the chance of a white marble is still about 55 percent (the average of 67 percent and 43 percent).
Greg Martin proved that for any number of marbles, say n white and n black marbles, it is best to put one white marble in one bag and the rest in the other. With the foe, the number of white marbles to put in the first bag is roughly twice the square root of n, minus 2.
Winners with best explanations: Reed Burkhart, Robert Lewis, Greg Martin.
New challenge question
(Thanks to Alice Loth.)
In the US, with our five coin denominations of 1, 5, 10, 25, and 50, it can take up to eight coins to make change (99). Which different five denominations would minimize the number of coins ever needed to make change?
*Send answers, comments, and new questions to:
Bronfman Science Center Williams College Williamstown, MA 01267
or by e-mail to:
The best submissions will receive a copy of the book "Flatland," which helps explain higher dimensions. Repeat winners will earn credits toward other prizes. Challenge question answers and winners will be announced in the next column in two weeks.