A bewildering puzzle by Lewis Carroll: Place 24 pigs in these sties so that, no matter how many times one circles the sties, he always find that the number in each sty is closer to 10 than the number in the previous one.
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Arrange the pigs as above. Now:
- 8 is closer to 10 than 6.
- 10 is closer to 10 than 8.
- Nothing is closer to 10 than 10.
- 6 is closer to 10 than 0.
Simple enough!
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Dec 30, 2012 | Categories: Puzzles
By Francis C. Collins. White to mate in two moves.
Dec 27, 2012 | Categories: Puzzles
A puzzle by Harry Langman:
A thin belt is stretched around three pulleys, each of which is 2 feet in diameter. The distances between the centers of the pulleys are 6 feet, 9 feet, and 13 feet. How long is the belt?
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This has a surprisingly neat solution. The belt can be divided into straight and curved sections. The lengths of the straight sections are the same as the distances between the centers of the pulleys, and the curved sections total one complete circumference. So the total length is 6 + 9 + 13 + 6.2832 = 34.2832 feet.
From Scripta Mathematica, March 1949.
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Dec 23, 2012 | Categories: Puzzles
By Francis Healey. White to mate in two moves.
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1. Qe8! and White will mate with 1. Qxe7, 1. Qh8, or 1. Nd7.
From A Collection of Two Hundred Chess Problems, 1866.
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Dec 19, 2012 | Categories: Puzzles
By John Brown. White to mate in two moves.
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With 1. Qg1 White sets up four pretty mates:
1. Qg1 Kd5 2. Qc5#
1. Qg1 Kd6 2. Qd4#
1. Qg1 Kf4 2. Bc7#
1. Qg1 f4 2. Qc5#
From Chess Strategy: A Collection of the Most Beautiful Chess Problems, 1865.
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Dec 13, 2012 | Categories: Puzzles
Helen Fouché Gaines’ 1956 textbook Cryptanalysis: A Study of Ciphers and Their Solution concludes with a cipher that, she says, “nobody has ever been able to decrypt”:
VQBUP PVSPG GFPNU EDOKD XHEWT IYCLK XRZAP VUFSA WEMUX GPNIV QJMNJ JNIZY KBPNF RRHTB WWNUQ JAJGJ FHADQ LQMFL XRGGW UGWVZ GKFBC MPXKE KQCQQ LBODO QJVEL.
It was still unsolved in 1968, when Dmitri Borgmann, editor of the Journal of Recreational Linguistics, urged his readers to tackle the problem: “Are you going to let this challenge lie there, taunting you for the rest of your lives? Or are you going to get busy and solve that pesky little crypt?”
So far as I can tell, they let it lie there, and it remains unsolved to this day. Any ideas? There are few clues in Gaines’ book. The cipher is the last in a series of exercises at the end of a chapter titled “Investigating the Unknown Cipher,” and she gives no hint as to its source. Of the exercises, she writes, “There is none in which the system may not be learned through analysis, unless perhaps the final unnumbered cryptogram.” The solution says simply “Unsolved.”
Dec 12, 2012 | Categories: Puzzles
From a 1921 essay by A.A. Milne:
TERALBAY is not a word which one uses much in ordinary life. Rearrange the letters, however, and it becomes such a word. A friend — no, I can call him a friend no longer — a person gave me this collection of letters as I was going to bed and challenged me to make a proper word of it. He added that Lord Melbourne — this, he alleged, is a well-known historical fact — Lord Melbourne had given this word to Queen Victoria once, and it had kept her awake the whole night. After this, one could not be so disloyal as to solve it at once. For two hours or so, therefore, I merely toyed with it. Whenever I seemed to be getting warm I hurriedly thought of something else. This quixotic loyalty has been the undoing of me; my chances of a solution have slipped by, and I am beginning to fear that they will never return. While this is the case, the only word I can write about is TERALBAY.
The answer is not RATEABLY, or BAT-EARLY, which “ought to mean something, but it doesn’t.” Rudolf Flesch notes that TRAYABLE is not a word, and that, though TEARABLY appears in small type in Webster’s Unabridged, “it obviously won’t do.”
What’s the answer? There’s no trick — it’s an ordinary English word.
Dec 8, 2012 | Categories: Puzzles
By Charles F. Stubbs. White to mate in two moves.
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The surprising 1. Bg1! forces Black to capture a rook, and White mates with the light-squared bishop.
From Chess Problems, 1904.
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Dec 7, 2012 | Categories: Puzzles
By James Pierce. White to mate in two moves.
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The subtle 1. Ne2 leaves Black with no options:
1. Ne2 Kc6 2. Rb3#
1. Ne2 Ke6 2. Rf6#
1. Ne2 Ke4 2. Nf6#
1. Ne2 Kc4 2. Rc3#
From Chess Problems, 1873.
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Nov 29, 2012 | Categories: Puzzles
By Francis Healey. White to mate in two moves.
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The absurd-looking 1. Re7! wins in every variation.
From A Collection of Two Hundred Chess Problems, 1866.
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Nov 23, 2012 | Categories: Puzzles
Martin Gardner called this the proudest puzzle of his own devising. When the pieces on the left are rearranged as on the right, a hole appears in the center of the square. How is this possible?
“I haven’t the foggiest notion of how to succeed in inventing a good puzzle,” he told the College Mathematics Journal. “I don’t think psychologists understand much either about how mathematical discoveries are made. … The creative act is still a mystery.”
Nov 19, 2012 | Categories: Puzzles
By George E. Carpenter. White to mate in two moves.
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1. Qh3 Ke4 2. Rc4#
From Will H. Lyons, Chess-Nut Burrs: How They Are Formed and How to Open Them, 1886.
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Nov 15, 2012 | Categories: Puzzles
A puzzle by Polish mathematician Paul Vaderlind:
Andre Agassi and Boris Becker are playing tennis. Agassi wins the first set 6-3. If there were 5 service breaks in the set, did Becker serve the first game?
(Service changes with each new game in the 9-game set. A service break is a game won by the non-server.)
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If there were 9 games in the set, then one player served 4 games and one served 5. Suppose Agassi served 4 games and Becker had k service breaks. That would mean that Agassi won 4-k of the games in which he served, and 5-k of the games in which Becker served (because there were 5 service breaks altogether). Then Agassi would have won a total of (4-k) + (5-k) = 9-2k games, an odd number. But we know that Agassi won 6 games, so this is a contradiction. Hence Agassi must really have served 5 rather than 4 games, and therefore he served the first game.
From The Inquisitive Problem Solver, 2002.
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Nov 13, 2012 | Categories: Puzzles
By Werner Keym, from Die Schwalbe, 1979. What were the last moves by White and Black?
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Black’s king is in check, so White must just have moved his knight from b4 to a2. But what was Black’s move before that? He cannot have moved his king, as there’s no square from which it can have come. The only other possibility is that Black moved a piece that’s not now on the board — a piece that White’s a2 knight has just captured. What can that have been? Not a knight or a pawn, as there’s no square that either of these can have come from. Nor a queen or a rook, as those could only have moved from a1, where they would have been checking the White king at the end of White’s last move. The only remaining possibility is a bishop: Black has just moved a bishop from b1 to a2, and White has responded by capturing it with a knight from b4.
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Nov 9, 2012 | Categories: Puzzles
Two adjoining lakes are connected by a lock. The lakes differ by 2 meters in elevation. A boat can pass from the lower lake to the upper by passing through the lock gate, which is closed behind it; then water is added to the lock chamber until its level matches that of the upper lake, and the boat can pass out through the upper gate.
Now suppose two boats do this in succession. The first boat weighs 50 tons, the second only 5 tons. How much more water must be used to raise the small boat than the large one?
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None. The same amount of water is necessary to raise any boat.
One way to make this intuitive is to imagine that a 2-meter “slab” of water is inserted at the bottom of the lock chamber. This will raise a boat of any size to the level of the higher lake.
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Nov 7, 2012 | Categories: Puzzles
This scale balances a cup of water with a certain weight. Will the balance be upset if you put your finger in the water, if you’re careful not to touch the glass?
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Yes. The water imparts an upward force on your finger as you immerse it, so you must exert a downward force to overcome it. That force is transferred to the glass and then to the scale, which sinks.
From Yuri B. Chernyak, The Chicken From Minsk, 1995.
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Nov 2, 2012 | Categories: Puzzles
A curious puzzle by George Koltanowski, from America Salutes Comins Mansfield, 1983. “Who mates in 1?”
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White could mate in three different ways if it were his turn, but it can’t be, as there’s no legal move that Black can just have made. So Black plays Bxf6#.
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Nov 1, 2012 | Categories: Puzzles
A puzzle by Lewis Carroll:
A bag contains one counter, known to be either white or black. A white counter is put in, the bag shaken, and a counter drawn out, which proves to be white. What is now the chance of drawing a white counter?
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It seems at first that the chance must be 1/2, as we’ve simply added a white counter and then removed it again, returning the bag to its original state.
But this is an error. After the white counter is added, the bag contains either (a) 2 white counters or (b) 1 white counter and 1 black counter. These states are equally probable. The chance of drawing a white counter from bag (a) is 1; from bag (b) it’s 1/2. So when the white counter is drawn, the chance that the bag had contained (a) 2 white counters or (b) 1 white counter and 1 black counter are proportional to (a) 1/2 × 1 (b) 1/2 × 1/2, i.e., 2 to 1. So there’s a 2/3 chance that the remaining counter is white.
In other words, the white counter that we drew is twice as likely to have come from a white-white bag than from a white-black bag.
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Oct 31, 2012 | Categories: Puzzles
A puzzle by Henry Dudeney:
A lady is accustomed to buy from her greengrocer large bundles of asparagus, each twelve inches in circumference. The other day the man had no large bundles in stock, but handed her instead two small ones, each six inches in circumference. “That is the same thing,” she said, “and, of course, the price will be the same.” But the man insisted that the two bundles together contained more than the large one, and charged a few pence extra. Which was correct — the lady or the greengrocer?
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Both were wrong, and the lady was badly cheated. She only got half the quantity that would be contained in the large bundle, and therefore should have been charged half the original price. A circle with the circumference half that of another must have its area a quarter that of the other.
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Oct 29, 2012 | Categories: Puzzles
Raymond Smullyan presented this puzzle on the cover of his excellent 1979 book The Chess Mysteries of Sherlock Holmes. Black moved last. What was his move?
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There’s no legal move by which Black could have removed one of his own pieces from the board, so he must have moved the king. It’s illegal for two kings ever to occupy adjacent squares, so the king must have moved to a8 from a7. White’s previous move must have placed him in check there. White cannot have accomplished this with a bishop move, as the white bishop cannot have moved to its present position from another diagonal. The only other possibility is a discovered check by a knight:
So, in this position, White moved the knight from b6 to a8, discovering check, and Black captured the knight with his king.
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Oct 25, 2012 | Categories: Puzzles
You’ve just won a set of singles tennis. What’s the least number of times your racket can have struck the ball? Remember that if you miss the ball while serving, it’s a fault.
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Once. You swing and miss each time you serve, and your opponent commits a double fault each time he serves. Eventually you find yourself serving at 6-5 or 7-6 in the tiebreak, and then you heroically serve an ace.
From Dick Hess, All-Star Mathlete Puzzles, 2009.
UPDATE: A reader points out that a superlatively lazy player can win a set without ever hitting the ball, thanks to the hindrance rule:
26. HINDRANCE
If a player is hindered in playing the point by a deliberate act of the opponent(s), the player shall win the point.
“On the crucial tie-break point, your opponent dashes across the net and tackles you, awarding you the point and the set.”
(Thanks, Scott.)
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Oct 24, 2012 | Categories: Puzzles
The Renaissance mathematician Niccolò Tartaglia would use this bewildering riddle to assess neophytes in logic:
If half of 5 were 3, what would a third of 10 be?
What’s the answer?
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The question asks us to discover whatever strange factor could cause 5/2 to give the result 3, and to apply the same factor to 10/3. So:
5/2 : 3 = 10/3 : x
Solving for x gives the result 4.
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Oct 22, 2012 | Categories: Puzzles
A mother takes two strides to her daughter’s three. If they set out walking together, each starting with the right foot, when will they first step together with the left?
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Never. After the mother’s first four strides and the daughter’s first six, they will again step together with the right foot, and the cycle repeats. Nowhere in that interval do they step together with the left foot.
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Oct 21, 2012 | Categories: Puzzles
By M. Charosh, from the Fairy Chess Review, 1937. White to mate in zero moves.
Oct 19, 2012 | Categories: Puzzles
