Black and White

dawson chess puzzle

From T.R. Dawson, a logic problem posing as a chess puzzle. If pinned men do not check, how can White mate in two moves?

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The black rook on d4 pins the white rook on d6, so the black bishop on b6 is free to move, and this means that the white knight on c7 is pinned and not checking the black king. White plays 1. f3, threatening to take the e2 pawn with his queen. This would be mate — it’s no use blocking the queen’s check with the d4 rook, because this reverses the chain of pins — now the pinned d3 rook doesn’t check, so the d6 rook does, the b6 bishop doesn’t, and the c7 knight does, giving mate anyway! Likewise: 1. f3 Rd5 2. Qxe2# (same idea as above — 2. … Rb5 is ineffective) 1. f3 Bxa5 2. Kc8# 1. f3 Bxc7 2. Nxc7# 1. f3 Bxe8 2. Kxe8# 1. f3 Qxe7+ 2. Kxe7# 1. f3 Rd2 2. Bxd2# 1. f3 Rxd6+ 2. Qxd6# Dawson wrote, “The idea amounts to ‘discovering mate’ from c7.” He dedicated the puzzle to Eduard Birgfeld. From Caissa’s Wild Roses, 1935.

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November 5, 2016November 13, 2016 | Puzzles

Alternate Routes

How many pairs of prime numbers are there whose sum is 999?

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One. Any pair of numbers with an odd sum must have an even member, and there’s only one even prime. So there’s only one answer, 997 + 2 = 999.

November 3, 2016November 13, 2016 | Puzzles · Science & Math

The Donjon Keep Window

dudeney donjon keep window puzzle

In Henry Dudeney’s Canterbury Puzzles, Sir Hugh De Fortibus takes his chief builder to the walls of his donjon keep and points to a window there.

“Methinks,” he says, “yon window is square, and measures, on the inside, one foot every way, and is divided by the narrow bars into four lights, measuring half a foot on every side.”

“Of a truth that is so, Sir Hugh,” says the builder.

“Then I desire that another window be made higher up whose four sides shall also be each one foot, but it shall be divided by bars into eight lights, whose sides shall be all equal.”

Bewildered, the builder says, “Truly, Sir Hugh, I know not how it may be done.”

“By my halidame!” exclaims De Fortibus in pretended rage. “Let it be done forthwith. I trow thou art but a sorry craftsman, if thou canst not, forsooth, set such a window in a keep wall.”

How can it be done?

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“The illustration will show how this was to be done. It will be seen that if each side of the window measures one foot, then each of the eight triangular lights is six inches on every side.” “Of a truth, master builder,” said De Fortibus slyly to the architect, “I did not tell thee that the window must be square, as it is most certain it could never be.”

October 25, 2016November 6, 2016 | Puzzles

Black and White

billiards chess puzzle

Billiards chess is a variant of traditional chess in which the pieces carom off the sides of the board at right angles. In the diagram above, the white bishop at a2 controls the diagonal a2-g8 as in normal chess, but its attack also “bounces” from g8 to h7 and then back along the h7-b1 diagonal. Both bishops attack and move along these “bouncing” lines. How can White mate the black king in two moves?

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The king is already nearly corralled; White just has to arrange a final attack along the long dark diagonal. 1. Ka1 e2 2. Bb2 does the trick; if 1. … Ke5 then 2. Bg7#. By F.F.L. Alexander and E.P. White, from The Problemist, June 1932.

October 21, 2016October 30, 2016 | Puzzles

Speaking Volumes

A puzzle by A. Kozlov from the Soviet popular science magazine Kvant:

Watching a solar eclipse, a girl asks her father how much farther away is the sun than the moon.

He says, “As far as I remember, 387 times farther.”

She says, “Then I can figure out how much greater is the sun’s volume than the moon’s.”

He thinks about this and says, “I think maybe you can.” How can she do this?

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During a solar eclipse the moon covers the sun almost perfectly. This means that the ratio of the sun’s radius to the moon’s radius is the same as the ratio of these two bodies’ distance from Earth. So the sun’s volume is larger than the moon’s by a factor of about 387³.

October 20, 2016October 30, 2016 | Puzzles

Royal Descent

A puzzle from Stuart Collingwood’s Lewis Carroll Picture Book, 1899:

A captive Queen and her son and daughter were shut up in the top room of a very high tower. Outside their window was a pulley with a rope round it, and a basket fastened at each end of the rope of equal weight. They managed to escape with the help of this and a weight they found in the room, quite safely. It would have been dangerous for any of them to come down if they weighed more than 15 lbs. more than the contents of the lower basket, for they would do so too quick, and they also managed not to weigh less either.

The one basket coming down would naturally of course draw the other up.

The Queen weighed 195 lbs., daughter 105, son 90, and the weight 75.

How did they do it?

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The weight is sent down; the empty basket comes up. The son goes down; the weight comes up. The weight is taken out; the daughter goes down; the son comes up. The son gets out; the weight goes down; the empty basket comes up. The Queen goes down; the daughter and weight come up together; the daughter gets out. The weight goes down; empty basket comes up. The son goes down; the weight comes up. The daughter removes the weight and goes down; the son comes up. The son sends down weight; empty basket comes up. The son goes down; the weight comes up. The son gets out; the weight falls to the ground.

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October 20, 2016October 30, 2016 | Puzzles

Black and White

blumenthal chess problem

By Oskar Blumenthal. White to mate in two moves.

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1. Qh1! and Black will lose no matter which knight he captures. From Schweizerische Schachzeitung, 1907.

September 28, 2016October 9, 2016 | Puzzles

Black and White

dawson half-move chess puzzle

One more chess curiosity by T.R. Dawson: How can White mate in two half moves?

The answer is to play the first half of Bg1-f2, and the second half of Bf1-g2, thus getting the white bishop from g1 to g2 and giving mate.

A fair-minded reader might ask why Black can’t pull the same trick, transferring his bishop from b8 to b7 to block the check. The answer, Dawson argues, is that some of the constituent moves are illegal: Black can’t combine Bb8-c7 and Bc8-b7 because a bishop on c8 would put the white king in an unreal check on h3; and he can’t combine Bb8-a7 and Ba8-b7 because a8 is occupied.

From Caissa’s Fairy Tales (1947).

September 7, 2016September 6, 2016 | Puzzles

Crossing Guard

Suppose some 2n points in the plane are chosen so that no three are collinear, and then half are colored red and half blue. Will it always be possible to connect each red point with a blue one, in pairs, so that none of the connecting lines intersect?

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Yes. Since n is finite, there are a limited number of ways to make the pairings. And for each way of making them, the lengths of the n segments add up to some total amount. Now, if an arrangement contains an intersection, this total length can be reduced by an appeal to the triangle inequality, which says that the sum of the lengths of any two sides of a triangle must be greater than the third side. In the example above, if we reassign the connections as shown, the total length of the connecting segments is reduced, because c < a + b and f < d + e. Hence the arrangement with the minimum total length must necessarily be free of intersections. (A Putnam paper problem from 1979, via Ross Honsberger’s Mathematical Gems, 1985.) 09/04/2016 Reader Jacob Bandes-Storch shows one way to establish the pairings.

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September 3, 2016September 11, 2016 | Puzzles

Match Point

https://commons.wikimedia.org/wiki/File:Wedding_rings.jpg — Image: Wikimedia Commons

In all 50 states, it’s legal for second cousins to marry. So can third cousins, fourth cousins, and cousins of any higher degree.

Alex and Zelda are third cousins but cannot marry. Why? If they were not related, they would be perfectly eligible.

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Because they’re brother and sister. Their parents are second cousins who married. That makes Alex and Zelda both siblings (because they have the same parents) and third cousins (because they have the same great-great-grandparents). (Offered by Mark Astolfi in MIT Technology Review, September/October 2008.)

August 30, 2016September 11, 2016 | Puzzles