If you write out the numbers from 1 to 5000 in American English (e.g., THREE THOUSAND EIGHT HUNDRED SEVENTY-THREE), it turns out that only one of them has a unique number of characters. Which is it? Spaces and hyphens count as characters.

# Puzzles

# “The Peace Problem”

In 1984, at the height of the Cold War, Ukrainian chess journalist F.S. Bondarenko dedicated this puzzle to English chess editor A.J. Roycroft in the spirit of peace and goodwill.

White’s army is arranged as above. Add Black’s army (the standard complement of 8 pieces and 8 pawns) so that no piece of either color is under attack.

# Quickie

From *Crux Mathematicorum*, May 1998:

If *x* is *x*% of *y*, and *y* is *y*% of *z*, where *x*, *y*, and *z* are positive real numbers, what is *z*?

# Podcast Episode 86: Lateral Thinking Puzzles

Here are six new lateral thinking puzzles to test your wits and stump your friends — play along with us as we try to untangle some strange situations using only yes-or-no questions.

# All Hands on Deck?

A reader named Hamp Stevens sent this conundrum to Martin Gardner, who published it in his *Mathematical Magic Show* (1965). Can these 25 cards be arranged to form five poker hands, each of them a straight or better (that is, straight, flush, full house, four of a kind, straight flush, or royal flush)? If it’s possible, find the five hands; if it’s not, prove that it’s impossible.

“This ingenious puzzle is quickly solved if you go about it correctly,” Gardner wrote. “A single card is the key.”

# Black and White

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

# 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)?

# 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 2*n* + 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?

# 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.

# 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?