Table and Tumblers

This problem originated in Russia, according to various sources, but no one’s sure precisely where:

Before you is a square table that can rotate freely. In each corner is a deep well, at the bottom of which is a tumbler that’s either upright or inverted. You can’t see the tumblers, but you can reach into the wells to feel their positions.

Periodically the table rotates and stops at random. After each stop, you can feel two of the tumblers and turn over either, both, or neither. If all four of the tumblers are in the same state — all upright or all inverted — then a bell sounds. Otherwise the table rotates again and you make another “move.”

Can you guarantee to ring the bell in a finite number of moves? If so, how?

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A logic puzzle by Lewis Carroll, July 2, 1893. What conclusion can be drawn from these premises?

  1. All who neither dance on tight-ropes nor eat penny-buns are old.
  2. Pigs that are liable to giddiness are treated with respect.
  3. A wise balloonist takes an umbrella with him.
  4. No one ought to lunch in public who looks ridiculous and eats penny-buns.
  5. Young creatures who go up in balloons are liable to giddiness.
  6. Fat creatures who look ridiculous may lunch in public, provided that they do not dance on tight-ropes.
  7. No wise creatures dance on tight-ropes if liable to giddiness.
  8. A pig looks ridiculous carrying an umbrella.
  9. All who do not dance on tight-ropes and who are treated with respect are fat.
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Black and White

wurzburg mate in 2

Otto Wurzburg, first prize, eighth tourney, Pittsburgh Gazette-Times, June 10, 1917. White to mate in two.

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Stormy Weather

stormy weather puzzle - 1

Stephen Barr observes that a pitched roof receives less rain per unit area than level ground does. This seems to mean that rain that falls at a slant will be less wetting than rain that falls vertically. Why isn’t this so?

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Horse Sense

horse sense puzzle

From the U.K. Schools Mathematical Challenge, a multiple-choice competition for students ages 11-14:

Humphrey the horse at full stretch is hard to match. But that is just what you have to do: move one match to make another horse just like (i.e. congruent to) Humphrey. Which match must you move?

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Catbird Seat

catbird seat 1

A ladder is leaning against a tree. On the center rung is a pussycat. She must be a very determined pussycat, because she remains on that rung as we draw the foot of the ladder away from the tree until the ladder is lying flat on the ground. What path does the pussycat describe as she undergoes this indignity?

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The Footsoldier

fabel footsoldier chess puzzle - solution

By Karl Fabel, from Weltspiegel, 1946. White adds a pawn and mates in two moves.

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Cut and Thrust

barbeau paper cutting puzzle - 1

University of Toronto math professor Ed Barbeau can take a rectangular piece of paper and, using only a pair of scissors, produce the object pictured above. How?

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Black and White

loyd chess problem

By Sam Loyd. White to mate in two moves.

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A riddle from 1655:

What is likest to a cat in a hole?

A cat out of a hole.

See Poser.

Sound Sense


Here is a class of a dozen boys, who, being called up to give their names were photographed by the instantaneous process just as each one was commencing to pronounce his own name. The twelve names were Oom, Alden, Eastman, Alfred, Arthur, Luke, Fletcher, Matthew, Theodore, Richard, Shirmer, and Hisswald. Now it would not seem possible to be able to give the correct name to each of the twelve boys, but if you practice the list over to each one, you will find it not a difficult task to locate the proper name for every one of the boys.

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Four Puzzles

  1. You’re playing bridge. Each of four players is dealt 13 cards. You and your partner find that between you you hold all 13 cards of one suit. Is this more or less likely than that the two of you hold no cards of one suit?
  2. As he left a restaurant, a man gave the cashier a card bearing the number 102004180. The cashier charged him nothing. Why?
  3. How can you position a marble on the floor of an empty room so that I can’t hit it with a baseball?
  4. Thrice what number is twice that number?
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Crime Scene

mortimer chess puzzle

A tricky problem by Ernest Clement Mortimer. This position was reached after Black’s fourth move in a legal chess game. Can you reconstruct the game?

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Installment Plan

A traveler wants to stay at an inn for seven days. He has no money, but he has a gold chain with seven links. The innkeeper agrees to accept this in payment for the week’s stay, but the traveler is reluctant to part with all seven links at once. He prepares to cut the chain into seven pieces.

The innkeeper stops him. If the traveler is willing occasionally to accept change in the form of links previously paid, then they can work out a plan that minimizes damage to the chain and yet permits the traveler to pay only what he owes on each successive day. How many links must they cut?

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Squeeze Play

A rail one mile long is lying on the ground. If you push its ends closer together by a single foot, so that the distance between them is 5279 feet rather than 5280, how high an arc will the rail make?

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Black and White

black and white -- hassberg, new york post

By Eric M. Hassberg, New York Post, April 21, 1945. White to mate in two moves.

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Hands Up


There are 13 ways to draw four of a kind and 40 possible straight flushes.

Why then does a straight flush beat four of a kind?

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The Interloper


Consider the classic story of Henry, a backwoods villager watching a theatrical performance, who leaps to the stage to save the heroine from the clutches of the villain and a horrible death. Of course Henry is mistaken if he thinks he can save the actress. She is not in danger; there is nothing to save her from. But the character she portrays is in danger and does need saving. Can Henry help her, despite the fact that he doesn’t live in her world?

— Kendall L. Walton, “How Remote Are Fictional Worlds From the Real World?”, Journal of Aesthetics and Art Criticism, 1978

Boxing Day

Death row is overcrowded, so the warden proposes a radical solution. He places 100 boxes in a sealed room. Each contains a slip of paper bearing the name of one of the 100 prisoners on the row.

Each prisoner will enter the room by one door, open 50 boxes, and exit by another door. Unless every prisoner can discover his own name, all 100 will be executed.

The prisoners will be supervised by a guard. They cannot communicate with one another, and they must leave the room as they found it, but the group can prepare a plan in advance and post it on the wall of the room.

If they proceed at random, their chance of succeeding is 1/2100, or about 0.00000000000000000000000000000008. What should they do?

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Accordion Commute

A puzzle by David Wells:

Every day I take the subway from Startville to Endville. Today I arrived at the Startville station to find that my train was just departing. I caught the next train to Endville, where I left the station at exactly the same time as if I had caught the first train. How did I manage this? The two trains traveled at the same speed, and I myself did not have to rush to make up the lost time.

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Black and White

black and white

By Auguste D’Orville. White to mate in two moves.

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Black and White

calvi chess problem

By Calvi. White to mate in two moves.

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Sudden Death

sudden death chess problem

“Paul Morphy IV” submitted this puzzle to the Journal of Recreational Mathematics in 1974. White might capture Black’s queen at this point, but instead can choose to mate on the move. How?

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“A Desperate Woman”

A puzzle from Henry Dudeney:

Her eyes were filled with tears, her face was flushed with anger, and her expression was one of indignation at the brutal injury to which she had been subjected.

“You monster of cruelty!” she cried, “I have borne with you too long! The very foundations of my being you have injured. Day by day I have endured your tortures. When first we met your ease and polish attracted me, and when you became my own my friends envied me. Yet see what I have suffered for your sake! You offer every opposition to my advancing myself. Your understanding is far too small for a large soul like mine. My standing in society you have entirely ruined. Had we never met I might have walked in peace. Begone! We part for ever.”

There was a moment’s convulsive breathing, a grinding of teeth, and a quick sigh. It was all over between them. One supreme effort, and she cast it from her —

“‘Whom?’ perhaps the reader will ask. That is the question. I leave you to fill in the blank.”

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