You must participate in a three-way duel with two rivals. Each of you is given a pistol and unlimited ammunition. Unfortunately, you, Red, are the weakest shot — you hit your target only 1/3 of the time. Black is successful 2/3 of the time, and Gray hits everything he aims at.
It’s agreed that you will take turns: You’ll shoot first, then Black, then Gray, and you’ll continue in this order until one survivor remains. At whom should you shoot?
What is the sum of all the figures in the numbers from 1 to 1 million?
Hint: With the right technique, this can be done in the head.
In 1938, Samuel Isaac Krieger of Chicago claimed he had disproved Fermat’s last theorem. He said he’d found a positive integer greater than 2 for which 1324n + 731n = 1961n was true — but he refused to disclose it.
A New York Times reporter quickly showed that Krieger must be mistaken. How?
By Sam Loyd. In how few moves can White force Black to mate him?
The answer is to castle queenside:
Now every Black move is mate.
Each point in an infinite plane is colored either red or blue. Prove that there are two points of the same color that are exactly 1 meter apart.
The Schachfreund, edited by M. Alapin, gives the following amusing Chess Skit. A well-known chess master allowed weak opponents to make as many moves as they pleased during five minutes, as odds, before the beginning of a game, with the provision that they confined their moves to their own half of the board. At the end of the five minutes the game commenced, the odds-giver having the first move. During the five minutes one of them had played: [1. a4 2. Na3 3. h4 4. Nf3 5. d4 6. Nd2 7. Rh3 8. Nac4 9. Raa3 10. Ne4 11. Qd2 12. Rhf3 13. g3 14. Bh3 15. Qf4 16. Rae3], whereupon the odds-giver resigned without having made a single move, as he could not avoid mate in two.
— The British Chess Magazine, January 1899
In the 1890s an eminent Scot began to publish short popular science articles under an assumed name, for “the fun of seeing if he [could] make another reputation for himself.”
He succeeded, publishing three articles in the National Geographic before the secret leaked out.
The pseudonym was H.A. Largelamb. Who was the man?
A calisson is a flat French candy traditionally manufactured in the shape of two equilateral triangles joined along an edge. Suppose a quantity of these are packed randomly into a hexagonal container:
Each candy must take one of three orientations: east-west, northeast-southwest, or northwest-southeast.
As it happens, no matter how the candies are packed into the hexagon, an equal number will take each of these three orientations.
In the May 1989 issue of the American Mathematical Monthly, Guy David and Carlos Tomei demonstrated this with a beautifully intuitive “proof without words.” What had they seen?
You’re in a pitch-dark room. On a table before you are 12 pennies. You know that 5 are heads up and 7 are tails up, but you don’t know which are which. By moving and flipping the coins you must produce two piles with an equal number of heads in each pile. How can you do this without seeing the coins?
Draw three nonintersecting circles of different sizes, and bracket each pair of them with tangents. Each pair of tangents will intersect in a point, and these three points will always lie along a line.
On being shown this theorem, Cornell engineering professor John Edson Sweet paused and said, “Yes, that is perfectly self-evident.” What intuitive proof had he seen?
