Who’s Laughing Now?

A.A. Bennett offered this puzzle in the American Mathematical Monthly of May 1937:

A car with n (n > 2) passengers of different speeds of mental reaction passes through a tunnel and each passenger acquires unconsciously a smudge of soot upon his forehead. Suppose that each passenger

(1) laughs and continues to laugh as soon as and only so long as he sees a smudge upon the forehead of a fellow passenger;
(2) can see the foreheads of all his fellows;
(3) reasons correctly;
(4) will clean his own forehead when and only when his reasoning forces him to conclude that he has a smudge;
(5) knows that (1), (2), (3), and (4) hold for each of his fellows.

Show that each passenger will eventually wipe his own forehead.

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I don’t have Bennett’s solution, but here’s mine: Take the simplest case, with three people, A, B, and C. Each can see smudges on his friends’ foreheads, so all are laughing. A reasons: Suppose my own forehead is clean. Then B can see only one smudge, the one on C’s forehead. But B sees that C is laughing. This should prove to B that his own forehead must have a smudge, and he should wipe it off. He isn’t doing so. This must mean that this whole supposition — that my own forehead is clean — is false. With that realization, A wipes his own forehead. And at that point his supposition comes true, and B cleans his own forehead, following the reasoning given above. Now C sees that A and B are laughing, and that they both have clean foreheads. This tells C that his own forehead must be smudged, and he cleans it. So that’s the case with three people. Now start over and add D. D can think: If my own forehead were clean, all of the above reasoning would hold and A would clean his forehead. He isn’t doing so. That must mean that my own forehead is dirty. D cleans his forehead and everything else plays out as above. Then similarly you can add E, and F, and as many additional people as you like, following the same idea.

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December 1, 2009January 22, 2013 | Puzzles

Formal Speech

A puzzle by Isaac Asimov:

What word in the English language changes its pronunciation when it is capitalized?

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Asimov was thinking of polish, but there are many others: august, herb, job, lima, natal, nice, rainier, reading, rodeo, tangier … … and the delightful Mousehole, a village in Cornwall whose name is pronounced mau-zill — and which presumably contains mouseholes. Thanks to Wade and Dominic for the long list.

November 28, 2009January 22, 2013 | Language · Puzzles

The Mail Plane

A motorcyclist was sent by the post office to meet a plane at the airport.

The plane landed ahead of schedule, and its mail was taken toward the post office by horse. After half an hour the horseman met the motorcyclist on the road and gave him the mail.

The motorcyclist returned to the post office 20 minutes earlier than he was expected.

How many minutes early did the plane land?

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The motorcyclist would have taken 20 minutes to go from where he met the horseman to the airport and back. Thus he was 10 minutes from the airport when he met the horseman. These 10 minutes plus the 30 minutes the horseman had been riding before they met make 40 minutes the plane was ahead of schedule. From B.A. Kordemsky, The Moscow Puzzles, 1956.

November 19, 2009January 22, 2013 | Puzzles

A Geometry Problem

A poser from 1821:

Mathematicians affirm that of all bodies contained under the same superficies, a sphere is the most capacious: But they have never considered the amazing capaciousness of a body, the name of which is now required, of which it may be truly affirmed, that supposing its greatest length 9 inches, greatest breadth 4 inches, and greatest depth 3 inches, yet under these dimensions it contains a solid foot?

What is this body?

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A shoe. Don’t blame me. From Rational Amusements for Winter Evenings, by John Jackson, “private teacher of the mathematics.”

November 16, 2009January 22, 2013 | Puzzles

The Card Challenge

A problem posed by Harry Houdini: Given a piece of cardboard measuring 4″ × 2.5″, cut it so that a person can pass completely through it without tearing it.

Can it be done?

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Cut the card as shown and it can be drawn out into an endless chain, with an opening large enough for a man to pass it over his body. “A thin person could use a much smaller card.” From Houdini’s Paper Magic, 1922.

November 13, 2009January 22, 2013 | Puzzles

“A Crime Story”

From the American journal Scripta Mathematica:

An elementary school teacher in New York state had her purse stolen. The thief had to be Lilian, Judy, David, Theo, or Margaret. When questioned, each child made three statements:

Lilian:
(1) I didn’t take the purse.
(2) I have never in my life stolen anything.
(3) Theo did it.

Judy:
(4) I didn’t take the purse.
(5) My daddy is rich enough, and I have a purse of my own.
(6) Margaret knows who did it.

David:
(7) I didn’t take the purse.
(8) I didn’t know Margaret before I enrolled in this school.
(9) Theo did it.

Theo:
(10) I am not guilty.
(11) Margaret did it.
(12) Lillian is lying when she says I stole the purse.

Margaret:
(13) I didn’t take the teacher’s purse.
(14) Judy is guilty.
(15) David can vouch for me because he has known me since I was born.

Later, each child admitted that two of his statements were true and one was false. Assuming this is true, who stole the purse?

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Theo is innocent because he says so twice. Then (9) is a lie. Since (9) is a lie, (8) is true. Since (8) is true, (15) is a lie. Since (15) is a lie, (14) is true. Judy is the thief.

November 11, 2009January 22, 2013 | Puzzles

A Riding Tour

a riding tour

Is it possible to move the knight from a1 to h8, visiting every square of the chessboard once?

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No. A knight lands on light and dark squares alternately. If it begins on a dark square and makes 63 moves, then the last move must end on a light square. h8 is dark. Therefore the task is impossible. For the same reason, the task is also impossible to a rook.

November 8, 2009January 22, 2013 | Puzzles

Misc

Which is worth more, a pound of $10 gold pieces or half a pound of $20 gold pieces?
A kazoo costs $1 plus half its price. How much does it cost?
On its March 1961 cover, MAD Magazine pointed out that 1961 was the first “upside-up” year — the first year that reads the same upside down — since 1881. What will be the next such year?

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The first. A pound of gold is worth more than half a pound. $2. 6009.

November 5, 2009January 22, 2013 | Puzzles

The Stopped Clock

Andrea’s only timepiece is a clock that’s fixed to the wall. One day she forgets to wind it and it stops.

She travels across town to have dinner with a friend whose own clock is always correct. When she returns home, she makes a simple calculation and sets her own clock accurately.

How does she manage this without knowing the travel time between her house and her friend’s?

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Before she leaves the house, Andrea winds her own clock and sets it to an arbitrary time. Then she notes the correct time at her friend’s house both when she arrives and when she leaves. When she returns home she consults her own clock to see how much time the whole trip has taken, subtracts the period she spent at her friend’s house, and divides the result by two to learn the travel time in each direction. By adding this interval to the time she noted as she left her friend’s house, she can infer the current time and set her own clock.

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October 31, 2009August 11, 2013 | Puzzles

Dilemma

You’re given a choice between two gifts: $5 and $1,000. You can choose either, but a bystander will give you $1 million if you choose irrationally. Can you do it?