Puzzles
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
x3 + y3 = z4.
“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:
State House
A quickie from Peter Winkler’s Mathematical Puzzles, 2021: Can West Virginia be inscribed in a square? That is, is it possible to draw some square each of whose four sides is tangent to this shape?
Black and White
Hooper’s Paradox
William Hooper published the oddity in 1774. The rectangle at the top measures 10 units by 3, giving an area of 30. But its dissected pieces seem to produce two other rectangles, with areas 12 and 20. Where did the two extra units come from?
“Imitative Chess”
A puzzle by Henry Dudeney:
A chessboard was on the table with the pieces all set up for a game. So I asked Dr. Bates to play a game with the Major on these conditions: Whatever move Bates made throughout, with the white pieces, the Major must exactly imitate with the black, and Bates must give checkmate on the fourth move. As an experiment, Bates started off with 1. e4, and Rackford replied with 1. e5. Then Bates played 2. Qh5, and the Major had to reply with 2. Qh4. This gave me a good opportunity to explain that White cannot now play 3. QxQ, because it would be impossible for Black then to imitate the move. Neither could he play 3. Qxf7+, because Black cannot do the same thing, as he would have to get out of check. White must always make a move that Black can copy, until the checkmate is actually given on the fourth move.
“This puzzle caused great interest, and it was some time before somebody (I think it was Strangways) hit on a solution.”
