Crowd Control

A knight’s tour is a series of moves by a chess knight such that it visits each square on the chessboard once. The example above is a “closed” tour because it ends on the square where it started.

This inspired a puzzle posed by Martin Gardner. If we filled a standard chessboard with knights, one on each square, could all 64 of them move simultaneously? The closed knight’s tour shows that they could — they form a long conga line, with each knight vacating a square for the knight behind it to occupy.

Gardner asks: Could the same feat be accomplished on a 5 × 5 chessboard?

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The answer is no, and we can prove it by observing the board’s coloring. A 5 × 5 chessboard has 13 squares of one color and 12 of another. The nature of a knight’s move means that it always jumps from a square of one color to a square of another. So on a 5 × 5 chessboard, 13 knights would have to jump onto 12 squares, which is impossible. So the task can’t be done.

December 15, 2021December 14, 2021 | Puzzles

Podcast Episode 365: Lateral Thinking Puzzles

For this final episode of the Futility Closet podcast we have eight new lateral thinking puzzles — play along with us as we try to untangle some perplexing situations using yes-or-no questions.

Intro:

Sears used to sell houses by mail.

Many of Lewis Carroll’s characters were suggested by fireplace tiles in his Oxford study.

The sources for this week’s puzzles are below. In some cases we’ve included links to further information — these contain spoilers, so don’t click until you’ve listened to the episode:

Puzzle #1 is from Greg. Here are two links.

Puzzle #2 is from listener Diccon Hyatt, who sent this link.

Puzzle #3 is from listener Derek Christie, who sent this link.

Puzzle #4 is from listener Reuben van Selm.

Puzzle #5 is from listener Andy Brice.

Puzzle #6 is from listener Anne Joroch, who sent this link.

Puzzle #7 is from listener Steve Carter and his wife, Ami, inspired by an item in Jim Steinmeyer’s 2006 book The Glorious Deception.

Puzzle #8 is from Agnes Rogers’ 1953 book How Come? A Book of Riddles, sent to us by listener Jon Jerome.

You can listen using the player above, download this episode directly, or subscribe on Google Podcasts, on Apple Podcasts, or via the RSS feed at https://futilitycloset.libsyn.com/rss.

Many thanks to Doug Ross for providing the music for this whole ridiculous enterprise, and for being my brother.

If you have any questions or comments you can reach us at podcast@futilitycloset.com. Thanks for listening!

November 29, 2021November 28, 2021 | Podcast · Puzzles

Black and White

kling chess problem

By Joseph Kling. White to mate in two moves.

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1. Re6! forces the black king to move to f1, when 2. Rf3 is mate. From The Chess Euclid, 1849.

November 26, 2021November 25, 2021 | Puzzles

Black and White

bull chess problem

By T.P. Bull. White to mate in two moves.

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1. Qd8! and the queen will mate. From Bull’s Chess Problems, via Charles F. Wadworth, Unique Chess Problems, 1890.

November 4, 2021November 28, 2021 | Puzzles

“The Hammock Puzzle”

By Sam Loyd. What’s the least number of cords you’d have to cut to divide this hammock into two pieces?

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Twelve: From Loyd’s Cyclopedia of Puzzles, 1914.

October 27, 2021October 26, 2021 | Puzzles

Exercise

https://commons.wikimedia.org/wiki/File:Euclid_Tetrahedron_4.svg — Image: Wikimedia Commons

A problem from Peter Winkler’s excellent collection Mathematical Puzzles, 2021:

Four bugs live on the four vertices of a regular tetrahedron. One day each bug decides to go for a little walk on the tetrahedron’s surface. After the walk, two of the bugs have returned to their homes, but the other two find that they have switched vertices. Prove that there was some moment when all four bugs lay on the same plane.

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Call the bugs A, B, C, and D. In the initial position, when D looks at the other three bugs, she sees triangle ABC labeled clockwise, but in the final position she sees it labeled counterclockwise (or vice versa). Since the bugs move continuously, some position must arise during the walk in which either D is on the plane ABC or ABC do not determine a plane. In the latter case they’re collinear, which means they share a plane with any other point, including D.

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October 26, 2021October 25, 2021 | Puzzles

“The Frogs and Tumblers”

A puzzle by Henry Dudeney. Frogs sit on eight of these 64 tumblers so that no two occupy the same row, column, or diagonal. “The puzzle is this. Three of the frogs are supposed to jump from their present position to three vacant glasses, so that in their new relative positions still no two frogs shall be in a line. What are the jumps made?” The frogs may not exchange positions; each must jump to a glass that was not previously occupied.

(“But surely there must be scores of solutions?” “I shall be very glad if you can find them. I only know of one — or rather two, counting a reversal, which occurs in consequence of the position being symmetrical.”)

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“The frogs that jump are George in the third horizontal row; Chang, the artful-looking batrachian at the end of the fourth row; and Wilhelmina, the fair creature in the seventh row. George jumps downwards to the second tumbler in the seventh row; Chang, who can only leap short distances in consequence of chronic rheumatism, removes somewhat unwillingly to the glass just above him — the eighth in the third row; while Wilhelmina, with all the sprightliness of her youth and sex, performs the very creditable saltatory feat of leaping to the fourth tumbler in the fourth row. In their new positions it will be found that of the eight frogs no two are in a line vertically, horizontally, or diagonally.” (Henry E. Dudeney, “The Professor’s Puzzles,” Strand 12:12 [December 1896], 720-726; “The Professor’s Puzzles: Solutions,” Strand 13:1 [January 1897], 106-109.)

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October 19, 2021October 18, 2021 | Puzzles

Black and White

phelps chess problem

“The Heart,” by F.B. Phelps. White to mate in two moves.

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1. Bb5 (there’s a metaphor in here somewhere) and the queen will mate. From the Central New Jersey Herald, via Charles F. Wadworth, Unique Chess Problems, 1890.

October 15, 2021October 13, 2021 | Puzzles

Diplomacy

A memorable puzzle from the Russian science magazine Kvant:

How can a goat, a head of cabbage, two wolves, and a dog be transported across a river if it’s known that the wolf is ‘culinarily partial to’ goat and dog, the dog is ‘on bad terms with’ the goat, and the goat is ‘not indifferent to’ cabbage? There are only three seats in your boat, so you can take only two passengers — animal or vegetable — at a time.

(You can keep order within the boat.)

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Keep the goat in the boat with you for the first two trips, taking first the dog and then the cabbage across and leaving them on the opposite bank. Take the goat back with you and leave it on the riverbank by itself as you take the two wolves across. Leave them with the cabbage and return with the dog. Collect the goat and bring it and the dog to the farther shore. (By E. Chernyshov, via Quantum, May 1990.)

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October 14, 2021October 13, 2021 | Puzzles

Starting Funds

Three men play a game, agreeing that in each round the loser will double the money of each of the other two. After three rounds, each man has lost one time, and each man has $24. How much did each have at the start of the game?

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If we work backward, we can see that at each step there’s only one way in which the money can have been distributed: After three rounds, they have $24, $24, and $24. So after two rounds the amounts must have been $12, $12, and $48. Hence after one round the amounts must have been $6, $42, and $24. And thus originally the amounts were $39, $21, and $12. Under the rules as described, with each man losing once, these must be the amounts that they started with. Note that the total amount held by the group is always $72. From Edward Barbeau, Murray Klamkin, and William Moser, Five Hundred Mathematical Challenges, 1995.

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October 8, 2021October 7, 2021 | Puzzles