The Cornish Cliff Mystery

dudeney cliff puzzle

Police have chased two thieves to the Cornish coast. There they find that two sets of footprints depart the hard road and cross soft soil to the edge of a cliff, where they end. The cliff juts out 200 feet above sea-washed boulders. No one could survive a fall from this height, there is no way to descend the cliff, and there are no other footprints.

The police have proven that the footprints match those of the fleeing criminals. The small foot belongs to Marsh, who apparently takes relatively long strides, walking heavily on his heels. Lamson takes shorter strides, treading more on his toes and evidently following behind Marsh, as he sometimes treads over the smaller man’s footprints.

The two men did not walk to the cliff edge and then return to the road by walking backward in their own footprints — such precision over a course of 200 yards is impossible. Accordingly the inspector says he will report that the criminals, hopeless to escape justice, have hurled themselves to their death.

“Then you will make a fatal mistake,” says Henry Melville, a visiting member of the Puzzle Club. “The men are alive and in hiding in the district.” He proves to be right, but how did the men get away from the edge of the cliff?

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Mens et Manus

David Hagen offered this puzzle in MIT Technology Review in 2007. The MIT logo can be thought of as a slider puzzle. In the figure above, can you slide the tiles about so that the gray I can escape through the opening at top left?

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Point to Point

point to point 1

Here’s a triangle, ABC, and an arbitrary point, D, in its interior. How can we prove that AD + DB < AC + CB?

The fact seems obvious, but when the problem is presented on its own, outside of a textbook or some course of study, we have no hint as to what technique to use to prove it. Construct an equation? Apply the Pythagorean theorem?

“The issue is more serious than it first appears,” write Zbigniew Michalewicz and David B. Fogel in How to Solve It (2000). “We have given this very problem to many people, including undergraduate and graduate students, and even full professors in mathematics, engineering, or computer science. Fewer than five percent of them solved this problem within an hour, many of them required several hours, and we witnessed some failures as well.”

Here’s a dismaying hint: Michalewicz and Fogel found the problem in a math text for fifth graders in the United States. What’s the answer?

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Dividing the Spoils

Ten pirates have 100 gold pieces and want to divide them according to the law of the sea, which says that the spoils go to the strongest. So they arrange themselves from weakest to strongest, P1, P2, …, P10. But these are democratic pirates, so they ask the strongest pirate to make a proposal as to how to divide the loot. All 10 pirates will then vote on it. If at least 50 percent of them support the proposal, then they’ll enact it and that’s that. Otherwise the proposer will be thrown to the sharks.

All pirates value their lives more than gold, all are rational, they cannot cut the gold pieces into smaller pieces, and no pirate will agree to a side bargain to share pieces. What proposal should the strongest pirate make in order to get the most gold?

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The Shortest Road

shortest road - 1

We want to build a road between two cities, A and B, that are separated by a river. We can build a bridge, but it must be perpendicular to the river’s banks, as shown. Where along the river’s length should we place the bridge if we want to minimize the total length of the road?

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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 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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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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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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