No Tabs

William West noticed this inscription in an alehouse near Brighton. What does it mean?

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My brewers have sent their clerk, And I must pay my score; For if I trust my beer What shall I do for more? From West’s Tavern Anecdotes and Sayings, 1881.

April 3, 2013April 12, 2013 | Puzzles

Black and White

johnson chess problem

By Richard Wright Johnson. White to mate in two moves.

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The winning move is hard to find: 1. Qb5! leads to mate by 2. Rh8, 2. Qe8, or 2. Nf4. From The Westminster Papers, 1874.

March 29, 2013April 5, 2013 | Puzzles

Black and White

andrews chess problem

By Henry John Clinton Andrews. White to mate in two moves.

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1. Qg1! and the d4 rook will give mate. From The Chess Problem, 1887.

March 22, 2013March 29, 2013 | Puzzles

Digit Sums

Sort the numbers 0, 1, 2, …, 123456 into two sets. In one set put all the numbers who digits add to an even sum; in the other put those whose digits produce an odd sum. Which set is larger?

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The digit sum of each even number differs by 1 from that of the odd number that follows it, so they have opposite parity. Thus the score remains tied until we reach the last number, 123456, whose digits total 21. So the second set is larger than the first by a single member. (By Polish mathematician Paul Vaderlind.)

March 19, 2013March 29, 2013 | Puzzles

Black and White

kling chess problem

By Josef Kling. White to mate in two moves.

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White wins with the pretty 1. Bd5! The pawn will queen next move. From the Chess Player’s Chronicle, 1862.

March 15, 2013March 22, 2013 | Puzzles

Work Planning

A logic exercise by Lewis Carroll — what conclusion can be drawn from these premises?

I despise anything that cannot be used as a bridge.
Nothing that is worth writing an ode to would be an unwelcome gift to me.
A rainbow will not bear the weight of a wheelbarrow.
Whatever can be used as a bridge will bear the weight of a wheelbarrow.
I would not take, as a gift, a thing that I despise.

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Rainbows are not worth writing odes to.

March 14, 2013March 22, 2013 | Puzzles

Times Square

Prove that the product of four consecutive positive integers cannot be a perfect square.

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If n is the smallest of the four integers, then the product is (n)(n+1)(n+2)(n+3) = (n² + 3n)(n² + 3n + 2) = (n² + 3n + 1)² – 1 This can’t be a perfect square, because two positive squares cannot differ by 1. From Angela Dunn’s Mathematical Bafflers, via David Wells’ Penguin Book of Curious and Interesting Puzzles, 1992.

March 13, 2013March 22, 2013 | Puzzles · Science & Math

Word Ladders

On Christmas Day 1877, assailed by two young ladies with “nothing to do,” Lewis Carroll invented a new “form of verbal torture”: Presented with two words of the same length, the solver must convert one to the other by changing a single letter at a time, with each step producing a valid English word. For example, HEAD can be converted to TAIL in five steps:

HEAD
HEAL
TEAL
TELL
TALL
TAIL

Carroll called the new pastime Doublets and published it in Vanity Fair, which hailed it as “so entirely novel and withal so interesting, that … the Doublets may be expected to become an occupation to the full as amusing as the guessing of the Double Acrostics has already proved.”

In some puzzles the number of steps is specified. In Nabokov’s Pale Fire, the narrator describes a friend who was addicted to “word golf.” “He would interrupt the flow of a prismatic conversation to indulge in this particular pastime, and naturally it would have been boorish of me to refuse playing with him. Some of my records are: HATE-LOVE in three, LASS-MALE in four, and LIVE-DEAD in five (with LEND in the middle).” I’ve been able to solve the first two of these fairly easily, but not the last.

But even without such a constraint, some transformations require a surprising number of steps. Carroll found that 10 were required to turn BLUE into PINK, and in 1968 wordplay expert Dmitri Borgmann declared himself unable to convert ABOVE into BELOW at all.

In a computer study of 5,757 five-letter English words, Donald Knuth found that most could be connected to one another, but 671 could not. One of these, fittingly, was ALOOF. In the wider English language, what proportion of words are “aloof,” words that cannot be connected to any of their fellows? Is ALOOF itself one of these?

In 1917 Sam Loyd and Thomas Edison made this short, which plays with similar ideas. The goat at the end was animated by Willis O’Brien, who would bring King Kong to life 16 years later:

March 11, 2013September 28, 2019 | Language · Puzzles

Black and White

kockelkorn chess problem

By Carl Kockelkorn and Johannes Kohtz, 1876. White to mate in two moves.

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The key is to promote the pawn to a bishop: 1. f8=B Kxe8 2. Qc8#. Only a bishop will do — queening the pawn gives stalemate.

March 8, 2013March 15, 2013 | Puzzles

Ends and Means

Prove that if each point in the plane is colored red, yellow, or blue, a unit segment must exist whose endpoints are the same color.

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Set up two abutting equilateral triangles ABC and BCD with unit sides, as above, and suppose A is red. In order to avoid a unit segment with similarly colored endpoints, D must also be red. By the same reasoning, every point at this distance from A must also be red. But this describes a monochromatic circle, and we can always find two points on this circle that are 1 unit apart. Thus it’s impossible to avoid having a unit segment with ends of the same color. From Joseph Konhauser, Macalester College, St. Paul., Minn., in the Pi Mu Epsilon Journal, 1984, 668, via Ross Honsberger, Mathematical Chestnuts From Around the World, 2001.

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March 5, 2013March 15, 2013 | Puzzles