A puzzle by Yoshinao Katagiri: A boy and a girl played rock paper scissors 10 times. Altogether the boy played rock three times, scissors six times, and paper once, and the girl played rock twice, scissors four times, and paper four times (though, in each case, the order of these plays is unknown). There were no ties. Who won?

# Puzzles

# Fixing a Point

A problem proposed by Richard Hoshino and Sarah McCurdy for *Crux Mathematicorum*, September 2008:

Five points lie on a line. Here are the 10 distances between pairs of points, listed from smallest to largest:

2, 4, 5, 7, 8, *k*, 13, 15, 17, 19

What’s *k*?

# Black and White

By E. Carney Jr. White to mate in two moves.

# Enlightenment

You’re in a dark room. The only light comes from an old LED digital alarm clock with four seven-segment displays. The time is displayed in 24-hour format, HH:MM (no seconds), and the leading digit is blank if not used. How much time passes between the room’s darkest state and its lightest?

# Black and White

From *La Nuova Rivista degli Scacchi*. White to mate in two moves.

# Harmony

A problem from the January 1990 issue of *Quantum*: Forty-one rooks are placed on a 10 × 10 chessboard. Prove that some five of them don’t attack one another. (Two rooks attack one another if they occupy the same row or column.)

# Quickie

One other quick item from *Eureka*, the journal of the Cambridge University Mathematical Society:

In its 1947 problem drive, the society proposed the following problem:

*To find unequal positive integers x, y, z such that*

*x ^{3} + y^{3} = z^{4}.*

“Although there were some research students in Theory of Numbers among those who tried, not one person succeeded in solving it within the time, yet the solution is extremely simple.” What is it?

# A Plate of 1,000 Cookies

A puzzle by David B., a mathematician at the National Security Agency, from the agency’s May 2017 Puzzle Periodical:

Steve, Tony, and Bruce have a plate of 1,000 cookies to share. They decide to share them in the following way: beginning with Steve, each of them in turn takes as many cookies as he likes (they must take an integer amount, greater than or equal to 1), and then passes the plate clockwise (with Tony sitting to Steve’s left, and Bruce sitting to Tony’s left). Nobody wants to feel like he hogged too many cookies, so they all want to avoid being the player at the end who has taken the most cookies. Additionally, nobody wants to feel cheated by finishing with the fewest cookies. Finally, given that the previous two conditions are definitely met, or definitely cannot be met, each player would like to maximize the number of cookies he eats. The players’ objectives can be summarized as follows:

Objectives:

- Have one player who has eaten more cookies than you, and one player who has eaten fewer cookies than you.
- Eat as many cookies as possible.

Objective #1 takes infinite priority over Objective #2. Assuming that all players are perfectly rational, that they are all aware of each other’s rationality and objectives, and that they cannot communicate with each other in any way, how many cookies should Steve take to ensure he meets both objectives and how many cookies will Tony and Bruce take if Steve takes the winning amount?

# Evolution

I just ran across this anecdote by Jason Rosenhouse in *Notices of the American Mathematical Society*. In a middle-school algebra class Rosenhouse’s brother was given this problem:

There are some horses and chickens in a barn, fifty animals in all. Horses have four legs while chickens have two. If there are 130 legs in the barn, then how many horses and how many chickens are there?

The normal solution is straightforward, but Rosenhouse’s brother found an alternative that’s even easier: “You just tell the horses to stand on their hind legs. Now there are fifty animals each with two legs on the ground, accounting for one hundred legs. That means there are thirty legs in the air. Since every horse has two legs in the air, we find that there are fifteen horses, and therefore thirty-five chickens.”

(Jason Rosenhouse, “Book Review: Bicycle or Unicycle?: A Collection of Intriguing Mathematical Puzzles,” *Notices of the American Mathematical Society*, 67:9 [October 2020], 1382-1385.)

# “Fermat’s Last Theorem”

A puzzle by H.A. Thurston, from the April 1947 issue of *Eureka*, the journal of recreational mathematics published at Cambridge University: