Riddle me this: how can 68 + (3×1) = 70?

Dear Cecil:

A coworker and I are arguing about the following riddle. I hoped you could give me a hand. A magician must cross a bridge carrying three gold pieces. He weighs exactly 68 kilograms, and each piece of gold weighs one kilogram. The bridge can carry no more than 70 kilograms or it will break. How does he cross the bridge safely, without throwing or dragging the gold across?

The answer supplied with the puzzle says he juggles the gold pieces and therefore never weighs more than 70 kg, because one of the gold pieces is in the air at all times. I disagree — the magician and his gold pieces make up a system that weighs an average of 71 kg, whether he throws and catches them or just carries them. What do you think?

Cecil replies:

I think anybody carting around three kilos of gold (approximate value = $41,000) can probably afford to find himself a better bridge. Maybe not the answer you were looking for, but ain’t that always the way?

Cecil has given some thought to offering a rigorous mathematical analysis showing that the downward force exerted on the bridge by the magician and his gold is, as you rightly surmise, 71 kg on average. I realize technically you’re supposed to express force in terms of newtons rather than kilograms. I also realize I’m trying to communicate with the general public, much of which is, let’s face it, mathematically challenged. I recall a conversation many years ago with a student at Northwestern who was convinced her fellow students were dopes. “Do you realize,” she said indignantly, “that half the people in this place scored below the median?”

So let’s skip the math and keep it simple. Suppose we accept the proposition that a juggling magician weighs only 70 kg because “one of the gold pieces is in the air at all times.” This is equivalent to saying that the magician walks across the bridge with two gold pieces in his pockets and the third floating over his head.

Well, you did say he was a magician. But if we rule out the supernatural it’s obvious the magician has to support that third gold piece somehow, yes? (Please say yes. I want to believe there’s hope.) However he does it, he increases his weight by the weight of the gold. Stay off that bridge.

Let’s cross that bridge when we come to it

Dear Cecil:

In response to your column involving the magician (“M”) of 68 kg who was to carry three gold coins of 1 kg each across a bridge that could only hold 70 kg, please be advised that M could not juggle the coins across the bridge [as Cecil stated]. However, the reason is not because the system’s average mass is 71 kg. It is because each coin is 1 kg. Using 9.81 m/s2 for the earth’s gravitational acceleration … [mathematical analysis supposedly proving the preceding omitted].

In addition, for coins of any mass or number, there are at least four other ways for M to cross the bridge: (1) M and coins enter a low orbit, and M touches the bridge lightly in midorbit; (2) M and coins enter a low-altitude trajectory, and M touches the bridge lightly in midtrajectory; (3) M sends one coin into a high-altitude trajectory, walks the other two coins across, dumps them off bridge, lightly contacts falling coin, exits bridge, and reaches back and catches falling coin; and (4) M crosses bridge while accelerating coins downward [thereby propelling himself up], lessening net downward forces on bridge. [Six single-spaced pages of analysis plus 22 pages of supporting notes omitted.]

Dan. You ever think about doing volunteer work for your church?

Dear Cecil:

Your discussion of the magician attempting to cross the bridge reminded me of the truck driver with two tons of birds in a truck with a one-ton capacity. He needed to stop every so often and bang on the side of the truck to keep half the birds flying.

Send questions to Cecil via cecil@straightdope.com.

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