15 Puzzle

A problem from the 1999 Russian mathematical olympiad:

Show that the numbers from 1 to 15 can’t be divided into a group A of 13 numbers and a group B of 2 numbers so that the sum of the numbers in A equals the product of the numbers in B.

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Suppose this were possible, and let x and y be the numbers in B. Then: (1 + 2 + … + 15) – x – y = xy 120 = xy + x + y 121 = (x + 1)(y + 1) We know that x and y fall between 1 and 15, so the only possible solution is (x, y) = (10, 10). But x and y must be two different numbers. So the task is impossible.

July 2, 2013July 12, 2013 | Puzzles

Cold Case

Enigma, the official publication of the National Puzzlers’ League, published this item in the “Chat” column of its August 1916 issue:

“The police department of Lima, O., is greatly puzzled over a cryptic message received in connection with the robbery of a Western Ohio ticket agent. Here it is: WAS NVKVAFT BY AAKAT TXPXSCK UPBK TXPHN OHAY YBTX CPT MXHG WAE SXFP ZAV FZ ACK THERE FIRST TXLK WEEK WAYZA WITH THX.”

As far as I can tell, in the ensuing 97 years it has never been solved. Any ideas?

June 29, 2013June 28, 2013 | Puzzles

Black and White

challenger chess problem

By Alfred Clement Challenger. White to mate in two moves.

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White wins with the quiet 1. g5!: 1. … Kf5 2. Bc2# 1. … Kd5 2. Bf3# 1. … e5 2. Qf3# From the American Chess Magazine, 1897.

June 28, 2013July 5, 2013 | Puzzles

In a Word

carfax
n. a place where four roads meet

Traveling between country towns, you arrive at a lonely crossroads where some mischief-maker has uprooted the signpost and left it lying by the side of the road.

Without help, how can you choose the right road and continue your journey?

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You know the town you just came from — use that knowledge to orient the signpost.

June 27, 2013July 25, 2013 | Language · Puzzles

A Martian Census

A room contains more than one Martian. Each Martian has two hands, with at least one finger on each hand, and all Martians have the same number of fingers. Altogether there are between 200 and 300 Martian fingers in the room; if you knew the exact number, you could deduce the exact number of Martians. How many Martians are there, and how many fingers does each one have?

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The key is that the number of fingers can tell us uniquely the number of Martians. That’s a tall order. It eliminates, say, 246 fingers, because that’s too ambiguous: There might be 82 Martians with 3 fingers each, or 3 Martians with 82 fingers each, and so on. The only possibility that avoids this uncertainty is that the quantity of Martians and the quantity of fingers per Martian are expressed by the same number, and that this number is not composite. That means we’re looking for the square of a prime number, and the only such number in the range 200-300 is 289, or 17². So there are 17 Martians, each of which has 17 fingers. This is the only number between 200 and 300 that permits such a definite assertion.

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June 25, 2013July 5, 2013 | Puzzles

Black and White

Bachmann chess problem

By Ulrich Bachmann. White to mate in two moves.

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1. Rd6! and White will mate with the knight or the bishop. From Der Schachfreund, 1898.

June 21, 2013June 28, 2013 | Puzzles

Three by Three

three by three puzzle

What’s the ratio between the areas of the two triangles?

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Invert the smaller triangle and it becomes clear that the ratio is 4 to 1.

June 17, 2013June 28, 2013 | Puzzles

Well Traveled

In Lord Dunsany’s Fourth Book of Jorkens, a member of the Billiards Club observes a book called On the Other Side of the Sun and says, “On the other side of the sun. I wonder what’s there.”

Jorkens, to everyone’s surprise, says, “I have been there.”

Terbut challenges this, but Jorkens insists he was on the other side of the sun six months ago. Terbut knows perfectly well that Jorkens was at the club six months ago, so he wagers £5 that Jorkens is wrong. Jorkens accepts.

“You have witnesses, I suppose,” says Terbut.

“Oh, yes,” says Jorkens.

“My first witness will be the hall-porter,” says Terbut. “And yours?”

“I am only calling one witness,” says Jorkens.

“Went with you to the other side of the sun?” asks Terbut.

“Oh, yes,” says Jorkens. “Six months ago.”

“And who is he?” asks Terbut.

Whom did Jorkens call?

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The Astronomer Royal. Six months ago we were all on the other side of the sun.

June 15, 2013June 21, 2013 | Puzzles

Black and White

shinkman chess problem

By William Anthony Shinkman. White to mate in two moves.

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After 1. Qh8! Black must move his knight, allowing 2. e6#. (If 1. … Nd4 or 1. … Ng7, then 2. Qa8#.) From the Ottawa Citizen, 1902.

June 14, 2013June 21, 2013 | Puzzles

Alcohol Problem

A bottle of fine wine normally improves with age for a while, but then goes bad. Consider, however, a bottle of EverBetter Wine, which continues to get better forever. When should we drink it?

— John L. Pollock, “How Do You Maximize Expectation Value?”, Noûs, September 1983

See The Devil’s Game.

June 11, 2013June 11, 2013 | Puzzles