Bread Alone

Andy and Bill are traveling when they meet Carl. Andy has 5 loaves of bread and Bill has 3; Carl has none and asks to share theirs, promising to pay them 8 gold pieces when they reach the next town.

They agree and divide the bread equally among them. When they reach the next town, Carl offers 5 gold pieces to Andy and 3 to Bill.

“Excuse me,” says Andy. “That’s not equitable.” He proposes another arrangement, which, on consideration, Bill and Carl agree is correct and fair.

How do they divide the 8 gold pieces?

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Assume each of the eight loaves was divided into 3 pieces, making 24 altogether. Andy’s 5 loaves made 15 pieces. He ate 8 and gave the remaining 7 to Carl. Bill’s 3 loaves made 9 pieces. He ate 8 and gave the remaining 1 to Carl. So Andy should get 7 gold pieces and Bill 1.

December 25, 2007January 16, 2013 | Puzzles

Three-Step

three step dudeney chess problem

Henry Dudeney says this puzzle is “supposed to be Chinese, many hundreds of years old, and never fails to interest.” White to play and mate, moving each of the three pieces exactly once.

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1. Rd6 Kc8 2. Ka7 Kc7 3. Rac6 mate

December 20, 2007January 16, 2013 | Puzzles

A Hiking Puzzle

Suppose a man sets out to climb a mountain at sunrise, arriving at the top at sunset. He sleeps at the top and descends the following day, traveling somewhat more quickly downhill. Prove that there’s a point on the path that he will pass at the same time on both days.

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The simplest solution is to imagine that the man’s twin climbs the mountain while he himself descends it. Clearly the two will meet at the same point, and at the same time.

December 10, 2007January 16, 2013 | Puzzles

A Warm Question

A puzzle from Henry Dudeney’s Amusements in Mathematics (1917):

There is a certain village in Japan, situated in a very low valley, and yet the sun is nearer to the inhabitants every noon, by 3,000 miles and upwards, than when he either rises or sets to these people. In what part of the country is the village situated?

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The sun is always closer when it’s overhead. The planet’s rotation carries us up to 4,000 miles closer between dawn and noon each day.

December 5, 2007January 16, 2013 | Puzzles

Tableau

Place two white rooks and a white knight on this board so that Black is checkmated:

The solution need not be reachable in a normal game.

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From Henry Dudeney, who ascribes it to Sam Loyd.

November 15, 2007January 16, 2013 | Puzzles

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