Black and White

carpenter chess problem

By George Edward Carpenter. White to mate in two moves.

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1. Qh3! Ke4 2. Rc4# From the Dubuque Chess Journal, 1873.

December 17, 2015December 27, 2015 | Puzzles

Hard Target

In a special football game, a team scores 7 points for a touchdown and 3 points for a field goal. What’s the largest mathematically unreachable number of points that a team can score (in an infinitely long game)?

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Eleven. For any integer n greater than that, the possible remainders when n is divided by 3 are 0, 1, and 2. If the remainder is 0, then we can get to n by scoring field goals. If the remainder is 1, then n – 13 has remainder 0 and we can reach n by scoring one touchdown and some number of field goals. And if the remainder is 2, then n – 14 has remainder 0 and we can get to n by scoring two touchdowns and some number of field goals. Put simply, all integers n greater than 11 can be written in the form n = 7a + 3b for some nonnegative integers a and b.

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December 16, 2015December 27, 2015 | Puzzles

Pleased, I’m Sure

A problem by Atlantic College mathematician Paul Belcher:

Anna and Bert invite n other couples to a dinner party. Before the meal begins, some people shake hands. No one shakes hands with their own partner, no one shakes hands with themselves, and no two people shake hands with each other more than once. Afterward, Anna asks all the other 2n + 1 people how many times they shook hands, and she gets a different answer from each of them. How many times did Anna shake hands?

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Altogether there are 2n + 2 people at the party. None shakes hands with themselves or their partner, so the maximum number of handshakes that one person could have participated in is 2n, and the minimum is 0. Anna receives a different answer from each of the 2n + 1 people she asks, so the answers she gets must be 0, 1, 2, …, 2n. Let the person who shook hands 2n times be called Max and the person who shook hands 0 times be called Minie (making no assumptions about gender). Max must have shaken hands with everyone but his or her own partner. Since Max has shaken hands with Anna, Max cannot be Bert and Max’s partner must be Minie. Now remove Max and Minie and all their handshakes. This decreases by 1 the number of handshakes that each of the others participated in, and leaves us facing essentially the same problem but with n – 1 other couples. If we repeat this process, eventually we’re left with Anna and Bert as the only two people remaining. In that situation, neither of them shakes hands. And since one handshake disappeared as each of the other n couples was removed from the problem, both Anna and Bert have shaken hands n times. (Paul Belcher, “But What Colour Are Anna’s Eyes?”, Mathematical Gazette 87:509 [July 2003], 322-323.)

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December 15, 2015December 27, 2015 | Puzzles

A Christmas Challenge

Here’s a unique challenge for the holidays — one of the United Kingdom’s intelligence agencies, GCHQ, is distributing the puzzle above on its Christmas card this year. (See GCHQ’s website for details and a high-resolution grid.)

The puzzle is a nonogram: Each row and column bears a string of numbers that indicates the lengths of consecutive runs of black squares that will appear there when the grid has been completed. For example, “3 3” in the eighth row means that in the finished puzzle two shaded sections of 3 squares each will appear somewhere along its length. Some squares in the grid have already been shaded to get you started.

“By solving this first puzzle players will create an image that leads to a series of increasingly complex challenges,” notes the agency. “Once all stages have been unlocked and completed successfully, players are invited to submit their answer via a given GCHQ email address by 31 January 2016. The winner will then be drawn from all the successful entries and notified soon after.” The agency invites players to make a donation to the U.K.’s National Society for the Prevention of Cruelty to Children if they’ve enjoyed the puzzle.

(Thanks, Alex.)

02/08/2016 UPDATE: The answers have now been revealed — see the link at the bottom of this post.

December 12, 2015February 8, 2016 | Puzzles

Product Recall

A problem from the 2004 Harvard-MIT Math Tournament:

Zach chooses five numbers from the set {1, 2, 3, 4, 5, 6, 7} and tells their product to Claudia. She finds that this is not enough information to tell whether the sum of Zach’s numbers is even or odd. What is the product that Zach tells Claudia?

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420. Giving the product of the five chosen numbers is equivalent to giving the product of the two unchosen numbers. The only possible products that are produced by more than more than one pair of numbers are 12 (produced by {3, 4} and {2, 6}) and 6 (produced by {1, 6} and {2, 3}). In the second case, the sum of both pairs is odd, so the sum of the five chosen numbers would have to be odd too. So the first must be the case, and product of Zach’s five chosen numbers is $\displaystyle \frac{1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7}{12}=420.$ (From Titu Andreescu et al., 104 Number Theory Problems, 2007.)

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December 8, 2015December 20, 2015 | Puzzles · Science & Math

Black and White

mate on the move

White to play and mate on his first move.

(Don’t spend too much time on this — it’s a bit of a trick.)

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The white king and queen have changed places, which is impossible if White is playing “up” the board. So he must be playing down — in which case he can deliver mate by capturing the rook. “It is this sort of problem that gives non-orthodox chess a bad name!” writes Stephen Addison in The Book of Extraordinary Chess Problems (1989). “Problems have been composed along these lines in the past by both Smullyan and Loyd.”

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December 4, 2015December 13, 2015 | Puzzles

Breaking the News

In the December 2012 issue of 1 Across magazine, longtime crossword composer John Graham included a special instruction above one of his puzzles:

“I have 18dn of the 19; no 27, just 13 15; no 2 or 6 or 1dn 26 yet — plenty of 10, though I wouldn’t have chosen the timing.”

Solvers discovered that 18 down was CANCER and 19 across was OESOPHAGUS. The full message read:

“I have CANCER of the OESOPHAGUS; no CHEMOTHERAPY, just PALLIATIVE CARE; no NARCOTIC or STENT or MACMILLAN NURSE yet — plenty of MERRIMENT, though I wouldn’t have chosen the timing.”

The puzzle was reprinted as cryptic crossword No. 25,842 in the Guardian the following month.

“It seemed the natural thing to do somehow,” Graham said. “It just seemed right.” He died in November 2013, and the Guardian published a tribute crossword to remember him.

(Thanks, Anthony.)

December 3, 2015December 3, 2015 | Death · Puzzles

A Self-Enumerating Crossword

sallows crossword

Here’s a unique crossword puzzle by Lee Sallows. There are no clues — instead, each of the 12 entries must take the form [NUMBER](space)[LETTER](S), like so:

EIGHT BS
NINETEEN XS
ONE J

And so on. Can you complete the puzzle so that the finished grid presents an inventory of its own contents?

(A couple observations to get you started: Because the puzzle contains 12 entries, the solution will use only 12 letters. And one useful place to start is the shortest “down” entry, which is too short to be plural — it must be “ONE [LETTER]”.)

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(Lee Sallows, “A New Type of Crossword Puzzle,” American Mathematical Monthly 94:7 [August-September 1987], 666; Joseph D.E. Konhauser, Dan Velleman, and Stan Wagon, Which Way Did the Bicycle Go?: And Other Intriguing Mathematical Mysteries, 1996, Problem 162.)

November 27, 2015February 22, 2018 | Puzzles

A Long Visit

Your tedious nephews, Kerry and Kelly, are not honest, but they’re orderly. One of them lies Mondays, Tuesdays, and Wednesdays, and tells the truth on other days, and the other lies on Thursdays, Fridays, and Saturdays, and tells the truth on other days. At noon, they have the following conversation:

Kerry: I lie on Saturdays.

Kelly: I will lie tomorrow.

Kerry: I lie on Sundays.

On which day of the week does this conversation take place?

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No one lies on Sundays, so Kerry is lying today. That means it’s not Saturday, but rather Monday, Tuesday, or Wednesday. Kelly, who is telling the truth today, says he will lie tomorrow, so today must be Wednesday. Adapted from Crux Mathematicorum, May 1998.