By Alexander Yarosh. The position above was reached in a legal game, except that one piece has been knocked off the board. What was it?
A problem from the 2004 Moscow Mathematical Olympiad:
An arithmetic progression consists of integers. The sum of its first n terms is a power of two. Prove that n is also a power of two.
My wife and I walk up an ascending escalator. I climb 20 steps and reach the top in 60 seconds. My wife climbs 16 steps and reaches the top in 72 seconds. If the escalator broke tomorrow, how many steps would we have to climb?
Only one of these statements is true. Which is it?
A. All of the below
B. None of the below
C. One of the above
D. All of the above
E. None of the above
F. None of the above
A conundrum by French puzzle maven Pierre Berloquin:
Caroline leaves town driving at a constant speed. After some time she passes a milestone displaying a two-digit number. An hour later she passes a milestone displaying the same two digits but in reverse order. In another hour she passes a third milestone with the same two digits (in some order) separated by a zero.
What is the speed of Caroline’s car?
1. A friend gives you a bottle that contains seven olives. Two of them are green and five are black. He bets that if you remove three olives at random from the bottle, they’ll include a green one. Should you take the bet?
2. Agnes has a tin of olives. It originally contained both black and green ones, but someone has been eating them, so she’s not sure of the colors of the 14 olives that remain. She removes 7 at random and finds that they’re all green. If the odds of this happening were exactly 50-50, what are the colors of the remaining 7?
In 2000, the residents of Luppitt, East Devon, installed a granite bench decorated with a variety of puzzles and curiosities that “it is hoped will be practical and entertaining for most of the next millennium.”
Among the puzzles is this “railway maze,” contributed by Roger Penrose. Make your way from Start in the upper left to Finish in the lower right. The catch is that your train has no reverse gear — you must move continually forward, following the natural curve of the track and making no sharp turns.
Print out two copies of this pattern, cut them out, and fold each along the dotted lines, making two identical solids. Then fit these two pieces together to make a regular tetrahedron.
This sounds dead simple, but it stumped me for a while. See if you can do it. (There’s no trick — the task is just what it seems.)
In S.S. Van Dine’s The Bishop Murder Case (1929), someone is killing chessplayers and leaving a black bishop at each crime scene. The prime suspect is John Pardee, promoter of a chess opening called the Pardee Gambit, which he hopes to establish in master play. But Pardee kills himself, despondent after losing to Akiba Rubinstein at the Manhattan Chess Club. It turns out that the real killer was only using the chess angle to throw suspicion onto others.
Van Dine based Pardee on a real person, Isaac Leopold Rice, who sponsored numerous tournaments in which his Rice Gambit was the required opening. But practice showed that the best White could hope for was a draw, and the line was abandoned after World War I. In 1979 Larry Evans wrote, “One of the most heavily analyzed openings in history is now never played, interred in a footnote of the latest opening manual.”
In the book, investigators determine that Pardee had faced the position above against Rubinstein shortly before his suicide. White has just realized that Black has a forced win in four moves. How does Black play?
Nine wolves share a square enclosure at the zoo. Build two more square enclosures to give each wolf a pen of its own.
A conundrum by the late brilliant Japanese puzzle maven Nob Yoshigahara.
Lee Sallows writes, “You have to solve this yourself, otherwise you won’t see how beautiful it is.”
A cylindrical problem by Miodrag Petkovic: Imagine that the board is rolled into a cylinder, with the a-file adjoining the h-file. How can White mate in two moves?
Kepler’s second law holds that a line segment connecting an orbiting planet to its sun sweeps out equal areas in equal periods of time: In the diagram above, if the time intervals t are equal, then so are the areas A.
If gravity were turned off, would this still be true?
Imagine a 1000 x 1000 chessboard on which a white king and 499 black rooks are placed at random such that no rook threatens the king. And suppose the king goes bonkers and wants to kill himself. Can he reach a threatened square in a finite number of moves if Black is trying actively to avoid this?