Alex Bellos

A variation on the Monty Hall problem introduces a valuable “golden goat” that the contestant wants more than the car. Because the host randomly reveals an ordinary goat while avoiding the car, the revealed information changes the odds: sticking with the original door wins the golden goat two-thirds of the time, while switching wins only one-third. This reverses the usual Monty Hall strategy, where switching is optimal for winning the car.

Alex Bellos presents a variation on the Monty Hall problem. Instead of trying to win the car, the contestant wants to identify a secretly valuable goat while the host, unaware of its value, reveals an ordinary goat and offers the contestant the chance to switch. Readers are asked whether switching is the best strategy, with the solution to be provided separately.

Alex Bellos presents the solution to a 3×3 grid puzzle proposed by Alex Chui, a Tonbridge School pupil and the first person to win an International Mathematical Olympiad medal for seven consecutive years. Because each row and column must contain one factor each of 2, 3 and 5, the placements of the three prime factors can be chosen independently, with six arrangements for each. The puzzle therefore has 6 × 6 × 6 = 216 valid grids.

British student Alex Chui became the most successful contestant in International Mathematical Olympiad history after winning his seventh consecutive medal, including five golds, and scoring a perfect 42/42 in the latest competition. The article presents a puzzle suggested by Chui: counting the ways to fill a 3×3 grid with positive whole numbers so that every row and column has a product of 30. The solution is subsequently made available, while readers are asked to discuss the problem without spoilers.

A puzzle explores how much solar energy enters a room protected by both curtains and roller blinds. Using thermal equilibrium and a simplified model of radiative heat flow, the solution shows that 33% of the Sun’s energy reaches the room. The puzzle is part of Alex Bellos’s recurring Monday series, with the question supplied by Alan Williams-Key.

A physics puzzle asks what percentage of the Sun’s energy enters a room protected by both roller blinds and curtains. The apparent answer of 25% is incorrect because the blinds, curtains and sunlight must be considered as a system in thermal equilibrium; readers are invited to reason through the heat flow before the solution is revealed.

A coin-flipping puzzle asks two players to guess each other’s results and find a strategy that improves on the apparent 25% chance of both guessing correctly. By agreeing to announce their own coin flips—or alternatively to guess the opposite result—the players win whenever the flips match, raising the probability of winning to 50%. The puzzle was supplied by probability expert Henk Tijms.

A probability puzzle imagines two contestants in separate booths flipping fair coins and guessing the other person’s result. Although independent guessing appears to give a 25% chance of both being correct, contestants can agree on a strategy that improves their odds. Readers are invited to determine the strategy and winning probability, with a solution promised in an update.

The piece provides solutions to three puzzles about how information can mislead. A school-grade example shows how a cohort’s median can fall even when every existing pupil improves, while paired polling results demonstrate Simpson’s Paradox, in which aggregated data reverses the trends seen in separate groups. It also announces Edward Barrett as the winner of an Anguish Languish contest, featuring a phonetic rendering of “Mary Had a Little Lamb,” and includes other reader submissions inspired by current UK politics and well-known phrases.

Alex Bellos presents three puzzles about how data, statistics and language can mislead. Readers must devise a scenario in which a syllabus improves every pupil’s grades despite lowering the median, interpret two polls whose results produce a misleading conclusion about gender differences, and create amusing examples of Anguish Languish, a nonsense language based on similar-sounding English words. The piece also promotes Kit Yates’s book You Don’t Know What You’re Missing and offers a prize for the best Anguish Languish sentence.

The puzzle asks for the smallest number beginning with 4 such that moving the initial 4 to the end produces a number equal to one quarter of the original. Testing numbers of increasing length leads to 410256, since 410256 divided by 4 equals 102564. The puzzle is attributed to the 1983 Moscow Mathematical Olympiad.

A number puzzle asks readers to find the smallest number beginning with 4 such that moving the 4 to the end produces a number equal to one quarter of the original. Readers are given hints to test numbers of increasing length, with the solution scheduled for a later update. The puzzle originates from the 1983 Moscow Mathematical Olympiad.

Alex Bellos provides solutions to four chess-themed mathematical puzzles: proving that the number of players with an odd number of games is even, showing why a knight cannot visit every square of a chessboard and finish on the opposite corner, demonstrating a six-move pawn promotion-and-return sequence, and solving a four-knight swapping puzzle by modelling it as a train-shunting problem. The puzzles are associated with We Solve Problems, a charity offering free maths circles for UK secondary-school pupils.

Four chess-inspired mathematical puzzles challenge readers to prove that the number of players with an odd number of games is even, determine whether a knight can traverse an 8×8 board exactly once between opposite corners, find the shortest collaborative sequence for a pawn to be promoted and return to its starting square, and swap two pairs of knights on an unusual grid. The puzzles are associated with We Solve Problems, a charity running free mathematics circles for secondary-school pupils across the UK, and solutions are promised later.

Alex Bellos provides solutions to two pronunciation puzzles. The first asks readers to find letters whose names are homonyms not containing the corresponding letter; the answers are Q, U, I, C and K, spelling “QUICK.” The second asks for words spelled identically but pronounced differently, with solutions including alternate, appropriate, content, delegate, discount, entrance, invalid, minute, present, produce, refuse and upset. The piece also credits Gerry Reynolds and Ryan McCormack as puzzle sources and invites readers to suggest future puzzles.

Alex Bellos presents two pronunciation puzzles. The first asks readers to identify five English letters whose homonyms are differently spelled words that do not contain the corresponding letter; those letters form a common word. The second asks readers to solve 23 pairs of same-spelled words with different pronunciations, based on definitions ranging from “second option” and “switch back and forth” to “rubbish” and “surprise victory.” Solutions are linked, with readers asked not to post spoilers and invited to share their favourite homonyms.

Alex Bellos explains the solution to a puzzle involving two equally wide snakes of different lengths escaping through separate passages. Passage A uses a loop that is longer than the short snake but shorter than the long snake, causing the longer snake to block itself when it doubles back. Passage B uses a floor hole that the short snake cannot cross without falling into, while the longer snake can bridge it, assuming the snake has some rigidity.

A mathematics puzzle asks readers to design two fixed escape passages for two equally wide snakes—one long and one short—so that each passage allows only one of them to escape. The constraints rule out moving mechanisms and narrower-than-width passages, while allowing the snakes to wiggle. Readers are invited to submit solutions without spoilers, with the puzzle attributed to Kvantik Magazine.

Solutions are provided for three mathematical puzzles. Sending one ship with a success probability of P is shown to be better than sending two ships with probabilities of P/2 each; two apparently different oracles can be distinguished by asking whether they are answering truthfully; and a subtraction pattern of the form XXYZ − XYZW = XW necessarily has Z = 8 and W = 9, with several possible values for X and Y. The puzzles are adapted from Mathematical Puzzles and Curiosities.

Alex Bellos presents three recreational mathematics puzzles from Mathematical Puzzles and Curiosities by Tanya Khovanova, Ivo David and Yogev Shpilman. The challenges concern choosing between ships with different success probabilities, distinguishing two kinds of yes-or-no oracles, and identifying a numerical pattern in faulty subtraction. The piece invites readers to solve the problems and notes that solutions were later published.