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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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

Trial by Fire

You are grilling steaks for Genghis Khan. Your little grill can broil two steaks at a time, but Genghis is hungry and wants three. That’s a problem: It takes 4 minutes to grill each side of a steak, so you’ll spend 8 minutes grilling the first two steaks, then another 8 grilling the third. Sixteen minutes is a long time to keep a warlord waiting.

How can you improve your time while still cooking the steaks thoroughly? Genghis really likes his well done.

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Put two steaks on the grill. After 4 minutes, flip one and replace the other with an uncooked steak. At the 8-minute mark you’ll have one well-done steak and can finish cooking the two half-done ones that remain. Total cooking time: 12 minutes. Bonus tenuously related factoid: In 2003, researchers found that nearly 8 percent of the men in a large region of Asia have nearly identical Y chromosomes. Khan ruled the largest empire in history, and the researchers speculate that power permitted Khan and his male relatives to procreate prolifically, giving them 16 million descendants today.

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January 23, 2011January 26, 2013 | Puzzles

Do-Si-Do

do-si-do 4 x 3 chess puzzle

I’m not sure who originated this puzzle. Can the white queen force the black king onto square a1 of this 4 x 3 chessboard?

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Yes, she can: 1. Qa2+ Kb4 (if Kc3 then Qa4, Kb2, Qb4+, and continue from step 4) 2. Qc2 Ka3 3. Qc4 Kb2 4. Qb4+ Kc1 (if Kc2 then Qa3, Kb1, Qc3, Ka2, Qb4, Ka1) 5. Qc3+ Kb1 6. Qb3+ Kc1 7. Qb4 Kc2 8. Qa3 Kb1 9. Qc3 Ka2 10. Qb4 Ka1 11/11/2018 Wait, a reader points out that after 1. Qa2+ Kb4 2. Qc2 Ka3 3. Qc4 Kb2 4. Qb4+ Black has 4. … Qa2 and it’s not clear that White can succeed. (Thanks, David.)

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January 19, 2011November 11, 2018 | Puzzles

Round and Round

Does the top of a rolling wheel move faster than the bottom? In his Cyclopedia of Puzzles (1914), Sam Loyd calls this “an old problem which has created a considerable discussion in the mechanical world.”

The rolling wheel retains its shape; it will arrive at its destination as a connected unit. This seems to imply that all of its parts are moving at the same speed. Yet the point in contact with the ground is moving not at all, while the top continuously overtakes it. Surely, then, the top is moving faster? “There is just enough of the mathematical and mechanical element in the make-up of the problem,” writes Loyd, “to provoke discussions from such as are well-up on these subjects.”

His answer: “The top of a wheel progresses exactly as fast as the bottom.” And, being Sam Loyd, he adds a wrinkle: “If the question referred to a mark on the tire the answer would be different, for the top is the highest point of the wheel and cannot revolve, for if it revolves the hundredth part of an inch it ceases to be the top.”

A second vexed wheel riddle; William James and Lewis Carroll consider related questions; a train moves simultaneously east and west.

January 18, 2011January 17, 2011 | Puzzles

Meant to Be

In 2007 Gordon Bonnet was going through some genealogical records when he ran across a marriage record for Edward DeVere Stewart and Etta Grace Staggers.

“It was only when I was putting the names in my database that I noticed something odd about them. What is it?”

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Both are typed entirely with the left hand. From Word Ways: The Journal of Recreational Linguistics, February 2007.

January 12, 2011January 26, 2013 | Puzzles

Huddled

By Sam Loyd. White to move and mate in two.

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White needs to play a waiting move with his king. This will force the black queen to abandon her defense of the rook on g2, and White can capture it, giving mate. The trick is that not any move will do. Of the five squares available to the white king, four are vulnerable to a check by the black queen, and such a check would prevent White from mating on the next move. Only 1. Ke7 leaves the king safe from attack — now Black must move his queen and lose to 2. Qxg2#. (Moving the rook is no better: 1. Ke7 Rh2 2. Qxf1#.)

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January 10, 2011January 26, 2013 | Puzzles