“To My Pupil”

Lewis Carroll’s 1885 puzzle book A Tangled Tale bears an anonymous dedication:

Beloved Pupil! Tamed by thee,
Addish-, Subtrac-, Multiplica-tion,
Division, Fractions, Rule of Three,
Attest thy deft manipulation!
Then onward! Let the voice of Fame
From Age to Age repeat thy story,
Till thou has won thyself a name
Exceeding even Euclid's glory!

He was thinking of Edith Rix, a child-friend who went on to study mathematics at Cambridge. She might have divined without asking that the dedication was intended for her. How?

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Read the second letter of each line.

May 31, 2019 | Literature · Puzzles

Exercise

In his later years Lewis Carroll would while away sleepless hours by solving mathematical problems in his head. Eventually he published 72 of these as Pillow Problems (1893). “All of these problems I thought up in bed, solving them completely in my head, and I never wrote anything down until the next morning.” Can you solve this one?

Prove that 3 times the sum of 3 squares is also the sum of 4 squares.

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This follows because 3(x² + y² + z²) = (x + y + z)² + (y – z)² + (z – x)² + (x – y)².

May 23, 2019May 23, 2019 | Puzzles

Audition

In January 1942, a series of letters to the Daily Telegraph complained that the newspaper’s crossword was too easy — anyone might solve it in a matter of minutes, they said. Accordingly the chairman of a London club offered to donate £100 to charity if anyone could solve a given crossword in 12 minutes. Editor Arthur Watson arranged a competition in the paper’s Fleet Street newsroom, and five people won the competition (though the fastest was later disqualified for misspelling a word). Several were later hired to work at Bletchley Park breaking German military codes.

The Telegraph published the “time test” puzzle later that month, and presented it again in 2014, inviting readers to try to solve it in 12 minutes. “Could you have been a codebreaker at Bletchley Park?”

May 19, 2019 | Puzzles

Geography

Brad Taylor offered this baseball puzzle in the October 2001 issue of MIT Technology Review:

The Red Sox beat the Orioles 9 to 4 in 17 innings. Where was the game played?

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Baltimore. In order to win by more than 4 in extra innings a team must bat first. So the Red Sox were visiting the Orioles. (Thanks, Bruce.)

May 16, 2019May 15, 2019 | Puzzles

Black and White

I don’t know who came up with this, but I thought it was cute. Can White checkmate Black in two moves from this position?

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Yes. 1. Rb3 forces 1. … Ka5, and now 2. Ra3 is mate. From the r/chess subreddit.

May 15, 2019May 15, 2019 | Puzzles

Unwholesome

Prove that if n is a positive integer greater than 1, then

$\displaystyle 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}$

is not an integer.

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Find the least common multiple of 2, 3, …, n, and recast each fraction using this denominator. Now all the numerators will be even except for one term, the one whose original denominator had been the highest power of 2 that doesn’t exceed n. Thus the sum of the numerators is odd while the common denominator is even.

May 15, 2019May 14, 2019 | Puzzles

Red and Blue

A problem from the 1996-1997 Estonian Mathematical Olympiad:

A square tabletop measures 3n × 3n. Each unit square is either red or blue. Each red square that doesn’t lie at the edge of the table has exactly five blue squares among its eight neighbors. Each blue square that doesn’t lie at the edge of the table has exactly four red squares among its eight neighbors. How many squares of each color make up the tabletop?

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The tabletop measures 3n × 3n, so we can divide it evenly into n² 3 × 3 squares that together tile the surface completely. For each of these 3 × 3 squares: If the unit square at the center is red, there are 5 blue squares and hence 3 red squares adjoining it. If the unit square at the center is blue, there are 4 red squares and hence 4 blue squares adjoining it. That means that each of our 3 × 3 squares is made up of the same constituents: 5 blue squares and 4 red squares. So the 3n × 3n tabletop is made up of 5n² blue squares and 4n² red squares. It remains to show that such a coloring is possible. See the link below, pages 31-32, for a diagram. (From “The Olympiad Corner,” Crux Mathematicorum 29:1 [February 2003], 22-36.)

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May 12, 2019 | Puzzles

Alice Chess

alice chess

English chess enthusiast V.R. Parton invented this variant in 1953. The game is played on two boards, starting with the normal opening position on one board and a second board empty. The trick is that a piece that completes its move on one board “vanishes strangely off its board to appear suddenly on the other board, magically out of thin air!”

There are two stipulations: A move must be legal on the board on which it’s played, and the square to which the piece is transferred on the opposite board must be vacant. (This means that a piece can make a capture only on the board on which it stands.)

This makes things very confusing. Here’s a simple three-move game. White plays 1. e4, which means that his king pawn advances from e2, passes “through the looking glass,” and arrives at e4 on the second board. Black responds with 1. … d5, so his queen pawn arrives at d5 on the second board. White plays 2. Be2, meaning that his king’s bishop is transferred to e2 on the second board. And now Black makes the mistake of playing 2. … dxe4 — his pawn on d5 on the second board captures the white pawn there, which means the white pawn is removed from play and the black pawn finds itself on e4 on the first board. That’s a mistake, because White can play 3. Bb5:

alice chess - fool's mate

The white bishop returns to the first board, checking the black king. Strangely, this is checkmate: Black can’t capture the attacking piece or move out of danger, and he can’t interpose a piece — any piece that he moves on the first board will find itself on the second, and there are no pieces on the second board that could move to c6 or d7.

Parton wrote, “One board being as a looking-glass to the other, the resulting play is a game which has a character as fantastic perhaps as Alice’s own game in Through the Looking-Glass.”

Here’s an Alice chess problem by Udo Marks. How can White mate in two moves?

alice chess - marks puzzle

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Remarkably, White wins with a simple waiting move, 1. Kb1, returning his king to the first board where it’s out of harm’s way. Now any move that Black makes exposes him to some inescapable danger. For example, if he moves his b-pawn then White can move his knight to c6 on the first board, checking Black’s king. There is no safe vacant square on the second board to which the king might move, so the game is over. Black has 44 available moves, but all lead to checkmate.

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May 10, 2019 | Puzzles

Podcast Episode 247: Lateral Thinking Puzzles

Here are six new lateral thinking puzzles — play along with us as we try to untangle some perplexing situations using yes-or-no questions.

See full show notes …

May 6, 2019May 13, 2019 | Podcast · Puzzles

Another Puzzling Commute

A few weeks after his first confusing journey home from the train station, Smith again finishes work ahead of schedule and takes an early train home. This time he arrives at his suburban station half an hour early. Again, rather than wait for the chauffeur, he starts walking home. And as before, he meets his chauffeur on the road, who picks him up promptly and takes him home. How many minutes early do they reach the house this time?

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In the first puzzle we learned that Smith takes 55 minutes to walk a distance that the car can cover in 5 minutes. That means the car travels 11 times as fast as Smith. Today, if Smith has been walking for t minutes when he meets the chauffeur on the road, then he has spared the chauffeur an additional journey of t/11 minutes from that meeting point to the station. Since the driver would have arrived at the station at the normal meeting time after that t/11 final portion, 30 minutes after Smith had started walking, t + t/11 = 30. That means t = 27.5, and the total time saved, 2t/11, is five minutes. (From Edward J. Barbeau et al., Five Hundred Mathematical Challenges, 1995.)

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May 5, 2019May 11, 2019 | Puzzles