Recalling Yesterday

From P.M.H. Kendall and G.M. Thomas, Mathematical Puzzles for the Connoisseur, 1962:

I’ve just been reading Jules Verne’s Around the World in Eighty Days — you know, where Phileas Fogg lost a day on the way round. Our science master says that ships put it right nowadays by having a thing called a Universal Date Line in the Pacific. When you cross the line from East to West you put the calendar on a day; and when you cross it the other way you put the calendar back. What I want to know is, when Puck put a girdle round the Earth in forty minutes and presumably did the right thing on crossing the Date Line, why didn’t he get back on the day before he started — or the day after, according to which way round he went?

I asked the English master this and he got quite cross about it and said it was nothing to do with Shakespeare. But if you flew round the earth as quickly as Puck it would matter, wouldn’t it?

Wouldn’t it? Why doesn’t Puck lose a day?

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“It is quite true that anyone traveling across the Date Line would have to gain or lose a day. However, Puck travelled sufficiently fast to overtake the Sun and the time. When he passed the twelve o’clock midnight time he would also have to pass from one day to another. This latter change will always opposes the change made at the Date Line, causing Puck to arrive back the day he set out.”

March 23, 2011January 27, 2013 | Puzzles

Black and White

loyd american indian

By Sam Loyd. White to mate in two moves.

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1. Bf8! An outlandish key move that wins in every variation, e.g., 1. … Bxb2 2. Bxh6#. From the New York Sunday Herald, 1889.

March 17, 2011January 27, 2013 | Puzzles

The Annual Liars

Two brothers are scrupulously truthful, with one exception: Each lies about his birthday on his birthday.

On New Year’s Eve you ask what their birthdays are. The first says “Yesterday” and the second says “Tomorrow.”

On New Year’s Day you ask again what their birthdays are. Again the first says “Yesterday” and the second says “Tomorrow.”

What are their birthdays?

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Each brother lies only one day a year, so each must have given at least one truthful response. That means the first brother was born on 12/30 or 12/31 and the second on 1/1 or 1/2. Now, the first brother cannot have been born on 12/30, for that would mean he’d lied unaccountably on New Year’s Day. And the second brother cannot have been born on 1/2, for that would mean he’d lied unaccountably on New Year’s Eve. Thus their true birthdays are 12/31 and 1/1. From Pierre Berloquin, The Garden of the Sphinx, 1981.

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March 13, 2011January 27, 2013 | Puzzles

A Chess Maze

nelson chess maze

Harry L. Nelson offered this puzzle in the Journal of Recreational Mathematics in 1983. The black king’s favorite square is c8, but he finds it is under attack by a white pawn. In how few moves can he correct this problem and return to a peaceful c8? White never moves. The black king can capture white pieces, but he may not visit any square more than once and may not enter check.

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The task requires 32 moves: 1. Kd7 2. Kd6. 3. Kxd5 4. Ke4 5. Ke3. 6. Kf2 7. Kxg1 8. Kxh2 9. Kg3 10. Kf4 11. Kf5 12. Ke6 13. Kf7 14. Kxg7 15. Kf6 16. Ke5 17. Kd4 18. Kc5 19. Kb6 20. Ka5 21. Kxa4 22. Ka3 23. Kxb2 24. Kc2 25. Kd2 26. Kxe2 27. Kd3 28. Kc4 29. Kxb5 30. Kxa6 31. Kxb7 32. Kc8

March 3, 2011January 27, 2013 | Puzzles

A Thought Experiment

Suppose you have two identical bolts. Hold each by its head, engage the threads as shown, and revolve one about the other. Will this action pull the heads closer together or drive them farther apart?

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Neither. In effect one bolt turns clockwise, the other counterclockwise at the same rate, so the distance between the heads will remain unchanged.

February 18, 2011January 27, 2013 | Puzzles

Black and White

sam loyd mate in two

By Sam Loyd. White to mate in two moves.

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1. Qa2 and the queen will mate next move. From the Detroit Free Press, Nov. 19, 1881.

February 15, 2011January 27, 2013 | Puzzles

Black and White

Sam Loyd chess puzzle, Detroit Free Press, 01-27-1877

By Sam Loyd. White to mate in two moves.

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1. Qa1 and White will mate next move. From the Detroit Free Press, Jan. 27, 1877.

January 29, 2011January 27, 2013 | Puzzles

A Pretty Problem

In Longfellow’s novel Kavanagh, Mr. Churchill reads a word problem to his wife:

“In a lake the bud of a water-lily was observed, one span above the water, and when moved by the gentle breeze, it sunk in the water at two cubits’ distance. Required the depth of the water.”

“That is charming, but must be very difficult,” she says. “I could not answer it.”

Is it? If a span is 9 inches and a cubit is 18 inches, how deep is the water?

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At the water’s surface, the bud marks one vertex of a right triangle. If the depth of the water is x inches, then (x + 9)² = x² + 36². If my algebra is good, then x = 67.5 inches. Mr. Churchill was quoting the 12th-century Sanskrit text Lilavati, in which the problem first appeared, to show his wife that mathematics can be poetic: In a certain lake swarming with geese and cranes, The tip of a bud of lotus was seen one span above the water. Forced by the wind, it gradually moved, and was submerged at a distance of two cubits. O mathematician, tell quickly the depth of the water. “There is something divine in the science of numbers,” Churchill tells her. “Like God, it holds the sea in the hollow of its hand. It measures the earth; it weighs the stars; it illumines the universe; it is law, it is order, it is beauty. And yet we imagine–that is, most of us–that its highest end and culminating point is book-keeping by double entry. It is our way of teaching it that makes it so prosaic.” (Thanks, Shreevatsa.)

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January 28, 2011January 27, 2013 | Literature · Puzzles · Science & Math

Paperwork

For 25 years, Macalester College mathematician Joe Konhauser offered a “problem of the week” to his students. Here’s a sample, from the collection Which Way Did the Bicycle Go? (1996):

Fifteen sheets of paper of various sizes and shapes lie on a desktop, covering it completely. The sheets may overlap one another and may even hang over the edge of the desktop. Prove that five of the sheets can be removed so that the remaining ten sheets cover at least two-thirds of the desktop.

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“Imagine that the desktop was painted just before the paper was put on it. Then some of the sheets of paper will have picked up some paint when they were put on the desktop. More precisely, each sheet of paper will have picked up paint on it from any part of the desk that it covers that is not covered by any sheet lower in the pile. Since the desktop is completely covered by the paper, the total area of the paint on all the sheets of paper is exactly equal to the area of the desktop. If we remove the five sheets with the least paint on them, then clearly the total area of paint on the sheets remaining is at least two-thirds of the area of the desktop, and therefore at least two-thirds of the desktop is covered by the remaining sheets.”

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January 25, 2011January 26, 2013 | Puzzles

Even Steven

A Scholar traveyling, and having noe money, call’d at an Alehouse, and ask’d for a penny loafe, then gave his hostesse it againe, for a pot of ale; and having drunke it of, was going away. The woman demanded a penny of him. For what? saies he. Shee answers, for y^e ale. Quoth hee, I gave you y^e loafe for it. Then, said she, pay for y^e loafe. Quoth hee, had you it not againe? which put y^e woman to a non plus, that y^e scholar went free away.

— John Ashton, Humour, Wit, & Satire of the Seventeenth Century, 1883

January 24, 2011January 23, 2011 | Puzzles