An Alarming Paradox

In 1735, an anonymous “lover of mathematicks” offered the following conundrum:

“‘Tis certainly Matter of Fact, that three certain Travellers went on a Journey, in which, tho’ their Heads travelled full twelve Yards more than their Feet, yet they all return’d alive, with their Heads on.”

How is this possible?

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The “three travellers” are Francis Drake, Thomas Cavendish, and William Dampier, each of whom circled the globe. On such a journey, your head does indeed travel farther than your feet — about 37 feet farther, if you’re 6 feet tall and the globe is a perfect sphere.

November 6, 2007January 16, 2013 | Puzzles

The Appraisal

Suppose I show you two old coins. One is dated 51 B.C., and the other is marked George I. Which is authentic?

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Neither is authentic. No one in 51 B.C. could have foreseen the existence of Jesus of Nazareth, much less that he would arrive in exactly 51 years. And George would not have been called George I until it became necessary to distinguish him from George II (“Victoria” will not become “Victoria I” until we have a “Victoria II”). 02/27/2024 UPDATE: Reader Emilio Díaz notes that King Juan Carlos I of Spain opted for a regnal number though no distinction was necessary. A joke runs Hoy voy a ver a Juan Carlos primero, y a la reina después (“Today I’m going to see Juan Carlos first, and the queen later”). The practice varies among monarchies. (Thanks, Emilio.)

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October 24, 2007February 27, 2024 | Puzzles

A Puzzling Invitation

If you were a British soldier in Malta in the 19th century, you might receive this card from a local dram-shop:

a puzzling invitation

What does it mean?

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Just read it straight through: Here stop and spend a social hour In harmless mirth and fun. Let friendship reign, be just and kind, And evil speak of none. From William T. Dobson, Literary Frivolities, Fancies, Follies and Frolics, 1880. See also Requiescat In Pace.

October 9, 2007December 16, 2016 | Puzzles

“The Barrel Puzzle”

From Amusements in Mathematics by Henry Ernest Dudeney (1917):

The men in the illustration are disputing over the liquid contents of a barrel. What the particular liquid is it is impossible to say, for we are unable to look into the barrel; so we will call it water. One man says that the barrel is more than half full, while the other insists that it is not half full. What is their easiest way of settling the point? It is not necessary to use stick, string, or implement of any kind for measuring. I give this merely as one of the simplest possible examples of the value of ordinary sagacity in the solving of puzzles. What are apparently very difficult problems may frequently be solved in a similarly easy manner if we only use a little common sense.

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All that is necessary is to tilt the barrel as in Fig. 1, and if the edge of the surface of the water exactly touches the lip a at the same time that it touches the edge of the bottom b, it will be just half full. To be more exact, if the bottom is an inch or so from the ground, then we can allow for that, and the thickness of the bottom, at the top. If when the surface of the water reached the lip a it had risen to the point c in Fig. 2, then it would be more than half full. If, as in Fig. 3, some portion of the bottom were visible and the level of the water fell to the point d, then it would be less than half full. This method applies to all symmetrically constructed vessels.

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October 6, 2007January 16, 2013 | Puzzles

A Modest Proposal

Okay, I’ll ask three questions, and if you miss one I get your house. Fair enough? Here we go:

A clock strikes six in 5 seconds. How long does it take to strike twelve?
A bottle and its cork together cost $1.10. The bottle costs a dollar more than the cork. How much does the bottle cost?
A train leaves New York for Chicago at 90 mph. At the same time, a bus leaves Chicago for New York at 50 mph. Which is farther from New York when they meet?

Don’t be hasty — your house is on the line.

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1. A clock strikes six in 5 seconds. How long does it take to strike twelve? Answer: Not 10 seconds, but 11 — one second per interval. 2. A bottle and its cork together cost $1.10. The bottle costs a dollar more than the cork. How much does the bottle cost? Answer: Not $1.00, but $1.05. 3. A train leaves New York for Chicago at 90 mph. At the same time, a bus leaves Chicago for New York at 50 mph. Which is farther from New York when they meet? Answer: Neither — when they meet they’ll be equally distant from anywhere, n’est-ce pas? You’ll hear from my attorney.

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October 4, 2007May 20, 2013 | Puzzles

“The Famous ‘Wheel Question'”

This question was proposed in the Scientific American, in 1868: ‘How many revolutions upon its own axis, will a wheel make in rolling once around a fixed wheel of the same size?’

The question brought to the editor of that paper many replies all claiming to have solved it. Yet the replies were about equally divided as to the number of revolutions, one part claiming one revolution and the other two revolutions. So much interest was manifested in it that Munn & Co. published The Wheel, June, 1868. It contains 72 pages, giving many of the solutions, illustrated by many diagrams.

— Miscellaneous Notes and Queries, August 1889

So who’s right?

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The rolling wheel makes two revolutions. You can prove this to yourself using two coins with milled edges—hold one in place and roll the other around it.

October 2, 2007January 16, 2013 | Puzzles

A Physics Problem

You’re in a rowboat in a swimming pool, and you’re holding a cannonball. If you throw the ball into the pool, will the water level rise or fall?

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It will fall. When the cannonball is in the boat, it displaces its weight in water. When it’s in the pool, it displaces only its volume.

September 12, 2007January 16, 2013 | Puzzles · Science & Math

Avoiding a Scandal

Three beautiful women and their jealous husbands want to cross a river, but the boat will hold only two people at a time. How can they arrange the crossing if no woman is to remain with a man unless her husband is present?

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Mrs. A and Mrs. C cross Mrs. A returns Mrs. A and Mrs. B cross Mrs. B returns Mr. A and Mr. C cross Mr. A and Mrs. A return Mr. A and Mr. B cross Mrs. C returns Mrs. A and Mrs. B cross Mrs. A returns Mrs. A and Mrs. C cross Other solutions are possible, but all require at least 11 crossings. (The price of propriety!)

August 23, 2007January 16, 2013 | Puzzles

A Literary Knight’s Tour

The knight’s tour is a recreation familiar to chessplayers: Move a knight about an empty chessboard so as to visit each square exactly once.

On this board, each square contains a syllable. Collect them in the right order and you’ll compose a six-line quotation from Shakespeare. What is it?

(Hint: Start on e4, “to”.)

A Literary Knight's Tour

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The quotation is from King John, Act IV, Scene II, lines 11-16: To gild refined gold, to paint the lily, To throw a perfume on the violet, To smooth the ice, or add another hue Unto the rainbow, or with taper-light To seek the beauteous eye of heaven to garnish, Is wasteful and ridiculous excess. The puzzle is taken from Miscellaneous Notes and Queries, January 1889.

August 21, 2007January 16, 2013 | Literature · Puzzles · Quotations

The Mutilated Chessboard

Take an ordinary chessboard and cut off two diagonally opposite corners. Now: Is it possible to tile the remaining 62 squares with 31 dominoes?

This calls for inspiration rather than trial and error. Most people see the solution immediately or not at all.

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No, it can’t be done. Each domino covers two squares, a light one and a dark one. Therefore a board can be tiled perfectly only if it has an equal number of light and dark squares. The original board met this requirement, but we’ve removed two squares of the same color (diagonal corners). So the task is impossible.

July 17, 2007January 16, 2013 | Puzzles