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Martin Gardner: 35 puzzles worth knowing

35 puzzles from Martin Gardner, each explained in a minute and credited to the original work. Free on Savvy.

• Twelve coins, one fake, three weighings

  You have 12 coins that look identical, but one is counterfeit and weighs slightly differently — you don't know if it's heavier or lighter. Using a balance scale only three times, h…

  — Martin Gardner (popularizer), Mathematical Games column

• A judge promises a surprise execution — and logic can't rule it out

  A judge tells a prisoner he'll be hanged at noon on one weekday next week, and that he won't know which day until the executioner knocks. The prisoner reasons it can't be Friday (h…

  — Martin Gardner, The Unexpected Hanging and Other Mathematical Diversions

• Gardner's two children

  Mr. Smith has two children, and at least one of them is a boy. What's the probability that both are boys?

  — Martin Gardner, Mathematical Games: Problems Involving Questions of Probability and Ambiguity

• Ten prisoners, ten hats, and one guess each — save yourselves with silence

  Ten prisoners stand in a line, each seeing every hat in front of them but not their own or anyone behind them. Hats are red or blue, assigned in any pattern. Starting from the back…

  — Martin Gardner (popularizer), Induction puzzles

• Roll one coin around another — how many times does it spin?

  Hold one coin still on a table. Roll an identical coin around its rim, without slipping, until it returns to its starting position. How many full rotations does the moving coin mak…

  — Martin Gardner, Mathematical Carnival

• Bulgarian Solitaire

  Take 15 tokens and split them into any piles you like — say 10, 3 and 2. Now repeat this move forever: take one token from every pile, and gather all the tokens you removed into on…

  — Martin Gardner, Last Recreations: Hydras, Eggs, and Other Mathematical Mystifications

• Fold a paper strip into a hexagon, and it hides more faces than it should have

  Take a long strip of paper, fold it into a flat hexagon, and glue the ends. It looks like it has just two faces. But pinch three alternating corners together and fold it open from …

  — Martin Gardner, Flexagons

• An ant crawls along a rope that stretches faster than it can ever walk

  An ant starts at one end of a taut rope 1 km long, crawling forward at a steady 1 cm per second. But the whole rope also stretches uniformly, at 1 km per second, carrying the ant f…

  — Traditional; popularised by Martin Gardner, Ant on a rubber rope

• Martin Gardner's nontransitive dice

  Picture four unusually numbered dice, A, B, C, and D. They're built so that A beats B more often than not, B beats C more often than not, and C beats D more often than not - yet D …

  — Martin Gardner, Scientific American

• Can a square be sliced into smaller squares, no two the same size, with none left over?

  Cut a large square into smaller squares — every piece a square, no two the same size, and no wasted space. Mathematicians call this 'squaring the square,' and for decades no one co…

  — Brooks, Smith, Stone and Tutte; popularized by Martin Gardner, Squared square

• Seven odd little blocks fit together into a perfect cube in 240 different ways

  In 1933, Danish polymath Piet Hein was sitting in on a lecture about quantum physics when he sketched a set of seven pieces, each made from three or four unit cubes glued together …

  — Martin Gardner, Mathematical Games: Piet Hein's Puzzling Soma Cube

• A perfect predictor already filled the box — do you still have a real choice?

  A near-perfect predictor has already decided what to put in a hidden box, based on forecasting what you're about to do — and is right almost every time. In front of you sit that hi…

  — Martin Gardner, based on a problem devised by physicist William Newcomb, Mathematical Games (Scientific American)

• Two oddly numbered dice roll exactly like a normal pair

  A standard pair of dice, numbered 1-6 each, produces the classic bell-shaped spread of totals from 2 to 12. Can you find a different pair of six-sided dice, still only positive num…

  — Martin Gardner, Mathematical Games column

• Martin Gardner's paradox of the second ace

  A bridge player is dealt 13 cards and announces, 'I have an ace.' What's the probability she has at least one more ace? Now suppose instead she'd said, 'I have the ace of spades.' …

  — Martin Gardner, Scientific American, 'Mathematical Games'

• The warden knows who's pardoned. He won't say. But he'll say something.

  Three prisoners, A, B and C, are told one of them has been picked at random for pardon. Prisoner A asks the warden — who knows the answer but can't reveal it — to name one of the o…

  — Martin Gardner, Mathematical Games column

• A sultan's 100 daughters, and only one chance to pick the best dowry

  You'll meet 100 candidates one at a time, in random order, and see each one's value immediately. You must accept or reject on the spot, no going back. What strategy gives you the b…

  — Martin Gardner, Mathematical Games (Scientific American column)

• A warden's harmless-seeming answer secretly changes a prisoner's odds

  Three prisoners, A, B and C, are told one of them — chosen at random — will be pardoned. Prisoner A, knowing the warden can't reveal his own fate, asks instead to be told the name …

  — Martin Gardner, Mathematical Games

• A strip of paper folded into a hexagon hides more faces than you'd expect

  Take a long strip of paper, fold it into a ring of triangles, and glue the ends: you've built a hexaflexagon. Pinch three alternating corners together and it opens out flat again —…

  — Arthur H. Stone and colleagues; popularized by Martin Gardner, Flexagon

• A judge promises a surprise execution next week. Logic seems to rule out every day.

  A prisoner is told he'll be hanged at noon on one weekday next week, but that the hanging will come as a total surprise — he won't know the day until the executioner knocks. He rea…

  — Origin disputed (Sweden, 1940s); popularized by Martin Gardner, Unexpected Hanging Paradox

• A pencil-and-paper game guarantees an ending — but why, and after how many moves?

  In Sprouts, two players take turns joining dots with a curved line (never crossing another line, and no dot may ever get more than three line-ends), adding a fresh dot on every lin…

  — Martin Gardner, Scientific American (Mathematical Games column)

• The Blind Bartender's Four Glasses

  Four glasses sit at the corners of a square turntable, each either right-side up or upside down — but you're blindfolded. Each turn you may touch any two glasses and flip either, b…

  — Martin Gardner, Mathematical Games (Scientific American column)

• Two cubes, ten digits each, and every date of the month

  A desk calendar shows any day of the month, 01 through 31, using just the front faces of two cubes side by side. Each cube has only 6 faces, so between them they have 12 faces for …

  — Martin Gardner, Mathematical Games column

• He walked a mile south, a mile east, a mile north, and was back where he started

  An explorer walks one mile due south, turns and walks one mile due east, turns again and walks one mile due north — and finds himself exactly back at his starting point. He sees a …

  — Martin Gardner, Mathematical Games column

• Three boxes are all mislabeled — one draw tells you what's really in every one

  Three boxes hold, respectively, two black marbles, two white marbles, and one of each — but a prankster has swapped every label, so all three boxes are now guaranteed to be wrong. …

  — Martin Gardner, Mathematical Games column

• A spoonful swapped between wine and water — which glass 'wins'?

  Two glasses hold equal amounts of wine and water. Take a spoonful of wine and stir it into the water glass. Then take a spoonful of that new mixture and stir it back into the wine …

  — Martin Gardner (popularizer), Hexaflexagons and Other Mathematical Diversions

• The Magic Hexagon: One Honeycomb, Every Line Equal

  Arrange the numbers 1 through 19 into a honeycomb of nineteen small hexagons — rows of 3, 4, 5, 4 and 3 — so that all fifteen straight lines, in every one of the three directions, …

  — Martin Gardner, Mathematical Games (Scientific American column)

• Six Dots, Two Colors, No Triangle Allowed

  Draw six dots and connect every pair with a line. Two players take turns coloring in one uncolored line at a time, each using their own color. The first player to complete a triang…

  — Martin Gardner, Scientific American: Mathematical Games

• Cut a Shape Into Smaller Copies of Itself

  Some shapes have a strange property: you can slice them into several smaller pieces, and every single piece is a smaller copy of the original shape. An L-shaped tromino, for instan…

  — Martin Gardner, Scientific American: Mathematical Games

• Two Crossed Ladders in an Alley: How Wide Is It?

  Two ladders of different lengths are propped up across a narrow alley, each leaning from the base of one wall to somewhere up the opposite wall, crossing each other partway up. You…

  — Martin Gardner, Mathematical Circus

• A family has two children, one is a boy, what are the odds the other is too?

  Mr. Smith has two children, and at least one of them is a boy. What's the probability both children are boys? Most people confidently answer 1/2. Martin Gardner posed this exact pu…

  — Martin Gardner, Mathematical Games column, Scientific American

• Four simple rules, and entire computers emerge with no players at all

  Conway's Game of Life plays out on an infinite grid where each cell is alive or dead, and every generation is decided by simple rules based on how many living neighbours each cell …

  — Martin Gardner, introducing John Horton Conway's game, Mathematical Games column, Scientific American

• Shuffle this deck once and one thing about it stays perfectly predictable

  Arrange a deck in a strict repeating four-suit pattern, then have a spectator cut it once and riffle-shuffle it together with itself, just a single time. Deal the shuffled deck off…

  — Martin Gardner, Mathematical Games: The Gilbreath Principle

• A 1942 film's throwaway line about a rose hid an exact geometry problem

  In the 1942 film Mrs. Miniver, a character describes a rose bred to sit in color exactly between a red circle and a pink one: draw a circle, inscribe the largest equilateral triang…

  — Martin Gardner, Mathematical Games: The Mrs. Miniver Problem

• An assistant hands over four cards, and a magician instantly names the hidden fifth

  A spectator picks five cards at random from a shuffled deck. An assistant looks at them, then hands a magician four of the five, one at a time, face up, with no words exchanged. Th…

  — Martin Gardner, Mathematics, Magic and Mystery

• Twiddle Two Bolts Together — Which Way Do the Heads Move?

  Two identical bolts sit head to head with their threads meshed together like gears. Hold each bolt firmly so it can't spin in place, then twirl them around each other the way you'd…

  — Martin Gardner, My Best Mathematical and Logic Puzzles

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