50 Fun Facts About Logic Puzzles

A 100 kg load of potatoes that is 99% water dries out until it is 98% water; what does it weigh now?

50 kg is all that remains: the 1 kg of solid potato must now be 2% of the load rather than 1%, so the total weight has to halve.

In the film WarGames, which simple game does a computer play against itself to learn that nuclear war cannot be won?

Tic-tac-toe always ends level when both sides play perfectly, so the computer's endless draws teach it that some games have no winner.

A drawer holds loose black socks and blue socks; without looking, how few must you pull to guarantee a matching pair?

Three is enough: with only two colours, the third sock must match one of the first two, a rule mathematicians call the pigeonhole principle.

How many people must gather in a room before it is more likely than not that two share a birthday?

23 people are enough, because a group that size already contains 253 different pairs, and any one of those pairs could be the match.

If 5 machines take 5 minutes to make 5 widgets, how many minutes would 100 machines need to make 100 widgets?

5 minutes is right, because each machine takes five minutes to build one widget, so a hundred machines working side by side finish a hundred in the same time.

In Tversky and Kahneman's test about Linda, an outspoken graduate, which description is logically the most probable?

A bank teller is the only safe pick: adding a detail can never make a description more likely, yet most people asked chose the feminist version.

A fair coin has landed heads twenty times in a row; what is the chance that the next toss is also heads?

1 in 2 is the chance on every single toss, because a coin has no memory; believing that tails is now overdue is known as the gambler's fallacy.

What is the greatest number of pieces a round pizza can be divided into using only four straight cuts?

11 pieces are possible, because a cut that crosses every earlier cut adds as many pieces as its own number: the fourth adds four to the seven made by three.

In Minesweeper, what does a number shown on an opened square tell the player about the squares touching it?

How many hide mines is what the number reports, counting all eight neighbours, so each safe click is a small deduction and not a guess.

Water pouring puzzles, solved by tipping liquid between jugs of fixed sizes, are nicknamed after which action sequel?

Die Hard with a Vengeance has a 1995 scene built on the puzzle, in which jugs holding three and five units must be used to measure out exactly four.

In the classic puzzle of ferrying a wolf, a goat and a cabbage one by one, how few river crossings are needed?

Seven crossings do it, four forward and three back, and the trick is that the goat rides back across once so it is never left alone with the wolf or the cabbage.

Which number comes next in the sequence 1, 1, 2, 3, 5, 8, 13?

21 follows, as 8 plus 13, in a pattern named after Fibonacci, the Italian whose book of 1202 used it to count a growing population of rabbits.

The Monty Hall problem, a three-door brain teaser, takes its name from the first host of which American game show?

Let's Make a Deal inspired the puzzle, in which switching doors wins two times in three; Monty Hall later demonstrated it to a reporter with a set of car keys.

Which board game, devised by Anthony Pratt during wartime air raids, is won by ruling out suspects, weapons and rooms?

Cluedo, sold in North America as Clue, was first patented under the blunt name Murder; its title is a play on Ludo, a board game few Americans knew.

Which code-breaking board game, invented by an Israeli postmaster in 1970, scores every guess with small key pegs?

Mastermind is based on Bulls and Cows, a pencil-and-paper game that may date back a century, and it was shown to the big toy firms before a plastics company took it on.

What is the smallest number of colours that is always enough to colour any flat map so no bordering regions match?

Four colours always suffice, as Francis Guthrie guessed in 1852 while colouring a map of England's counties; the proof arrived only in 1976, with a computer's help.

What is the most queens that can stand on a standard chessboard with no queen able to attack any other?

Eight is the limit, one queen for each row, and the puzzle of arranging them, first published in 1848, has exactly 92 solutions.

Which society, founded at Oxford in 1946, admits anyone scoring in the top two percent on an approved IQ test?

Mensa takes its name from the Latin word for table, chosen to suggest a round table at which every member sits as an equal.

Which number comes next in the sequence 2, 3, 5, 7, 11, 13, 17, whose members divide only by themselves and one?

19 is the next prime, and the list never runs out: Euclid showed around 300 BC that there are infinitely many primes.

In which game-theory puzzle does each of two partners gain by betraying the other, though both do worse if both betray?

The prisoner's dilemma was devised at the RAND Corporation in 1950, and when two colleagues played a hundred rounds they often chose to cooperate after all.

Cards show 3, 8, blue and red; which two must you flip to test the rule 'even numbers have blue backs'?

8 and red are the only cards that could break the rule, yet fewer than one person in ten chose them in Peter Wason's original study.

Each term of the sequence 1, 11, 21, 1211, 111221 reads the previous term aloud; which term comes next?

312211 comes next, because the term before it reads as three 1s, two 2s, one 1; start from 22 instead and the sequence never changes at all.

Four walkers who need 1, 2, 5 and 8 minutes must cross a two-person bridge sharing one torch; how few minutes suffice?

15 minutes is the best possible, and the trick is to send the two slowest walkers over together, which beats the tempting plan of having the fastest escort everyone.

The classic puzzle of two uneven fuses that each burn for one minute asks solvers to time which exact interval?

45 seconds is the target: light both ends of one fuse and one end of the other, then light the last end when the first fuse dies after half a minute.

One of nine identical-looking coins is slightly light; how few weighings on a balance scale are sure to find it?

2 weighings always work: set three coins against three to learn which trio holds the light one, then compare two coins from that trio.

In a Tower of Hanoi, where no disc may sit on a smaller one, how few moves will shift five discs?

31 moves is the minimum, since each extra disc doubles the count and adds one; in the legend sold with the puzzle, priests moving 64 golden discs will finish as the world ends.

Why can 31 dominoes never tile a chessboard that has had two diagonally opposite corners cut away?

Both missing squares share a colour, which leaves 32 squares of one colour and 30 of the other, while every domino has to cover one of each.

In the classic letter sum SEND + MORE = MONEY, where each letter hides a digit, which digit must M be?

1 is forced, because adding two four-digit numbers can carry no more than one into a fifth column; Henry Dudeney printed the puzzle in The Strand Magazine in 1924.

On an island of constant liars and constant truth-tellers, a native says 'my friend and I are both liars'; what follows?

Only the speaker lies: no truth-teller could call himself a liar, so the claim is false, and it can be false only if the friend is honest.

Three children's ages multiply to 36 and add up to 13, and only one child is the eldest; what are the ages?

2, 2 and 9 is the only fit: 1, 6 and 6 also makes 36 and 13, but then the eldest would be a twin, the clue that settles the classic census-taker puzzle.

In the Zebra Puzzle that Life International printed in 1962, which of the five house-owners turns out to drink water?

The Norwegian drinks water and the Japanese owner keeps the zebra; none of the puzzle's clues mentions water or a zebra, so both answers have to be deduced.

What is the smallest number of starting clues a standard Sudoku grid can have while still having only one solution?

17 clues is the floor: an exhaustive computer search published in 2014 proved that no Sudoku with 16 clues has a single solution.

Researchers using Google's computers proved in 2010 that any scrambled Rubik's Cube can be solved within how many moves?

20 moves is the ceiling, a figure nicknamed God's number, and millions of scrambled positions genuinely need all twenty.

Which grid puzzle, also sold as Picross or Paint by Numbers, reveals a hidden picture from number clues along its edges?

Nonogram joins the name of Non Ishida, the Japanese designer who got the idea after making pictures from lit skyscraper windows, to the word diagram.

Which arithmetic grid puzzle did maths teacher Tetsuya Miyamoto invent in 2004 to train the brain without instruction?

KenKen takes its name from the Japanese word for cleverness; publishers without rights to the trademark sell the same puzzle as Calcudoku or Mathdoku.

Which puzzle maker offered $1,000 for solving a 15 puzzle with tiles 14 and 15 swapped, a feat that is impossible?

Sam Loyd could safely offer the prize, since the swap had been proved impossible years earlier; he also claimed to have invented the 15 puzzle, which he had not.

In a famous tour puzzle once studied by Leonhard Euler, which chess piece must land on every square exactly once?

The knight makes the tour, and a route that finishes one knight's move from where it began, ready to go round again, is called a closed tour.

Leonhard Euler proved in 1736 that no walk could cross each of the seven bridges of which Prussian city exactly once?

Königsberg, now Kaliningrad in Russia, gave mathematics a new field: Euler's proof that the walk is impossible laid the foundations of graph theory.

Which gentleman archaeologist solves villagers' brain teasers with his apprentice Luke in a hit Nintendo DS series?

Professor Layton made his debut in the Curious Village, and in every game the townsfolk he meets keep handing him fresh puzzles and mysteries.

Which Parade columnist drew some 10,000 letters, most calling her wrong, for correctly solving the Monty Hall problem?

Marilyn vos Savant, once listed by Guinness for the highest recorded IQ, was right to say switch doors, though nearly 1,000 of the letter writers held PhDs.

Connect Four was solved in 1988: the first player can always force a win by dropping the opening disc where?

The middle column is the only winning start; open one column to either side and perfect play gives a draw, while a start at the edge actually loses.

Which writer ran the Mathematical Games column in Scientific American for 25 years, popularising recreational logic?

Martin Gardner wrote paper-folding puzzles for a children's magazine before the column, which grew out of an article of his on folded-paper hexaflexagons.

A judge promises a condemned man a surprise hanging on a weekday next week; which day does the man rule out first?

Friday goes first, since a hanging on the last possible day could be no surprise; he then strikes out every other day the same way, and is caught off guard midweek.

In George Boolos's Hardest Logic Puzzle Ever, one god always lies and one never does; how does the third god answer?

At random is how the third god replies, and to make matters worse all three answer in their own language, using da and ja for yes and no in an unknown order.

In the islanders puzzle, 100 blue-eyed logicians must leave once sure of their eye colour; on which dawn do they go?

The 100th dawn after a visitor mentions seeing blue eyes empties the island: each islander sees 99 others, and only when none has left after 99 dawns is each sure of being one more.

With Chicken McNuggets sold only in boxes of 6, 9 and 20, what is the largest number you cannot buy exactly?

43 is the largest total no mix of boxes can reach; the puzzle's author, Henri Picciotto, worked it out on a napkin while eating at McDonald's with his son.

Three boxes hold two gold coins, two silver, and one of each; having drawn gold, how likely is the other coin gold?

2 in 3 is right, because three gold coins could have been drawn and two of them lie in the all-gold box; Joseph Bertrand set the trap in a book of 1889.

Which logician, also a magician and concert pianist, wrote popular puzzle books about truth-tellers and liars?

Raymond Smullyan gave one 1978 collection the teasing title What Is the Name of This Book?, and said his introduction to logic was an April Fools' Day trick played by his older brother.

Linking three houses to gas, water and power with no crossed lines is impossible on paper but possible on which surface?

A doughnut, or torus, works because one line can slip through the hole; that is why the puzzle is sometimes printed on a coffee mug, whose handle does the same job.

In a finished standard Sudoku, where every row, column and box holds each digit once, how often does the digit 7 appear?

9 times is the answer, once in each of the nine rows; the puzzle's name is short for a Japanese phrase meaning that the digits must stay single.