“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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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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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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Black and White

gold chess problem

Henry Dudeney in Strand, June 1911: “It would be difficult to find a prettier little chess problem in three moves, produced from such limited material as a rook and a pawn, than the one given this month, by Dr. S. Gold. The novice will probably find the task of discovering the key move quite perplexing. White plays and checkmates in three moves.”

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The Switchback Puzzle


“Some years ago there was a craze for rolling pellet puzzles,” wrote Henry Dudeney in 1909, “though they are really more trials of patience than puzzles.”

One exception was this undulated glass tube, which contained three shots or pellets. The task was to get them into the three depressions at A, B, and C, which are unfortunately positioned at high points in the tube.

This “could be solved by a puzzle trick which I was surprised to notice how few people discovered,” Dudeney wrote. What was it?

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Made to Order

Arrange the digits 0-9 into a 10-digit number such that the leftmost n digits comprise a number divisible by n. For example, if the number is ABCDEFGHIJ, the number ABC must be divisible by 3, ABCDE must be divisible by 5, and so on.

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Black and White

ensor chess puzzle

From Henry Dudeney’s Perplexities column, Strand, March 1911:

“Here is a pretty little chess puzzle, made some years ago by Mr. F. S. Ensor. White has to checkmate the Black king without ever moving a queen off the bottom row, on which they at present stand. It is not difficult. As the White king is not needed in this puzzle, His Majesty’s attendance is dispensed with. His three wives can dispose of the enemy without assistance — in seven moves.”

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Exam Week

A problem submitted by the United States and shortlisted for the 16th International Mathematical Olympiad, Erfurt-Berlin, July 1974:

Alice, Betty, and Carol took the same series of examinations. There was one grade of A, one grade of B, and one grade of C for each examination, where A, B, C are different positive integers. The final test scores were

Alice: 20
Betty: 10
Carol: 9

If Betty placed first in the arithmetic examination, who placed second in the spelling examination?

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