Two Olive Problems

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?

Railway Mazes

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.

Foursquare

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.)

Black and White

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?

Black and White

By Dezso Elekes. White to mate in two moves.

Black and White

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

Separate Quarters

Nine wolves share a square enclosure at the zoo. Build two more square enclosures to give each wolf a pen of its own.

Nob’s Number Puzzle

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.”

Black and White

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?

Black and White

By John Griswold White. White to mate in two moves.

No Attraction

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?

Black and White

By Sam Loyd. White to mate in two moves.

Math Notes

Prove that log10 5 is irrational.

Black and White

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

The Suicidal King

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?

Black and White

By Hyacinth R. Agnel. White to mate in two moves.

Black and White

By William Anthony Shinkman. White to mate in two moves.

Trivium

The names of two U.S. state capitals end with the same eight letters. What are they?

Black and White

By Sámuel Gold. White to mate in two moves.

“Anglo-Foreign Words”

Walter Penney of Greenbelt, Md., offered this poser in the August 1969 issue of Word Ways: The Journal of Recreational Linguistics. Below are five groups of English words. Each group appears also in a foreign language. What are the languages?

1. aloud, angel, hark, inner, lover, room, taken, wig
2. alas, atlas, into, manner, pore, tie, vain, valve
3. ail, ballot, enter, four, lent, lit, mire, taller
4. banjo, chosen, hippo, pure, same, share, tempo, tendon
5. ago, cur, dare, fur, limes, mane, probe, undo

Projective Chess

In projective geometry, every family of parallel straight lines intersects at an infinitely distant point. Chess problem composers in the former Yugoslavia have adapted this idea for the chessboard, adding four special squares “at infinity.”

Now a queen on a bare board, for example, can zoom off to the west (or east) and reach a square “at infinity” from which she attacks every rank on the board simultaneously from both directions. She might also zoom to the north (or south) to reach a different square at infinity; from this one she attacks every file simultaneously, again from both directions. Finally she can zoom to the northwest or southeast and attack all the diagonals parallel to a8-h1, or zoom to the northeast or southwest and attack all the diagonals parallel to a1-h8. These four “infinity squares,” plus the regular board, make up the field of play.

N. Petrovic created the problem below, published in Matematika Na Shahmatnoi Doske. White is to play and mate in at least two moves. Can you find the solution?

Reunion

A Russian problem from the 1999 Mathematical Olympiad:

A father wants to take his two sons to visit their grandmother, who lives 33 kilometers away. His motorcycle will cover 25 kilometers per hour if he rides alone, but the speed drops to 20 kph if he carries one passenger, and he cannot carry two. Each brother walks at 5 kph. Can the three of them reach grandmother’s house in 3 hours?

Black and White

By Ottmar Weiß. White to mate in two moves.