# Juggling in a Raft

Ask Marilyn ® by Marilyn vos Savant is a column in Parade Magazine, published by PARADE, 711 Third Avenue, New York, NY 10017, USA. According to Parade, Marilyn vos Savant is listed in the "Guinness Book of World Records Hall of Fame" for "Highest IQ."

## Question

Fred Moolten <fmoolten@acs.bu.edu> remembers the following Marilyn column. If you have this column, please send it to me.

A reader asked how a person could transport three balls across a river on a raft that could sustain a maximum weight of the traveller plus two of the balls. Marilyn assumed that multiple crossings were not permitted, although the puzzle did not state this. To her credit, this assumption converted an easy problem into a challenging one.

Marilyn considered a clever approach to the challenge: reduce the force of the balls on the raft. In considering how this might be done, she mentioned juggling the balls so that the traveller was not holding them all at once. She then explained that this would not work because the impact of a second ball descending into the hand of a juggler already holding one would bring the total downward force beyond the limits of buoyancy.

Although Marilyn did not discuss juggling all three balls so that there are always at least two in the air at a time, this would not work either. The problem with juggling as a solution does not depend on any need to handle two balls at once. Rather, for three balls to end a long journey at the same altitude as when they started, the upward force exerted during the trip must average their total weight, regardless of whether it is exerted constantly (by holding them), or intermittently (by juggling them). By Newton's law of action and reaction, this translates into an average downward force that also equals their weight.

## An alternative

If you can't juggle your way across the river, is there another weight reduction possibility? Wrap the balls in your shirt, and drag the package behind you in the water. As long as the density of the balls doesn't exceed three times the density of water, the effective weight of three of them in the water won't exceed the weight of two of them on the raft. (As an alternative, you could leave the balls on the raft and get in the water yourself - but watch out for crocodiles).

## A Restriction

Rod Schmidt <rschmidt@pierian.com> reports that wrapping the balls in your shirt and dragging them behind the raft will work if, and only if, the density of the balls does not exceed three times the density of the water. This is because the maximum weight of the three balls in the water must not exceed the weight of two balls in the raft. Since the weight of each ball in the water is reduced by the weight of the water it displaces, the three balls together must displace at least as much water as the weight of one ball. In other words, each ball must displace at least one third of its own weight in water. In order for a ball to displace one third of its weight in water, its density must not exceed three times the density of the water.

## Hey, You Can't Do That!

GE Wagner <gwagner@att.net> claims that the solutions of dragging the balls along behind are both incorrect because they removed the balls from the raft (the puzzle clearly asks "how a person could transport three balls across a river on a raft") and because their answers wouldn't work with balls of all compositions.

Removing the person from the raft is a nice idea because the puzzle doesn't say the raft must support the person. It just says it is capable of supporting the weight of the person and two balls. However, in a case where the balls each weighed more than the person, this solution would still not work.

I do however have a solution for you.

Since all three balls must be on the raft and since the solution must work no matter the comparative weight of ball to person, the only solution is to transport an additional item with you that lightens the overall load.

Just attach a big helium balloon to the raft. Make sure it's capable of offsetting the weight of one ball and enjoy your trip.

Editor's note: This solution assumes that there are no overhanging branches that would snag the balloon or the harness between balloon and raft.

http://www.wiskit.com/marilyn/raft.html last updated June 30, 1998 by herbw@wiskit.com