## Gear Trouble

A problem from the U.S.S.R. mathematical olympiad:

You’re given 13 gears. Each weighs an integral number of grams. Any 12 of them can be placed on a pan balance, with 6 in each pan, so that the scale is in equilibrium. Prove that all the gears must be of equal weight.

## Coming and Going

A puzzle by Pierre Berloquin:

In my house are a number of rooms. (A hall separated from the rest of the house by one or more doors counts as a room.) Each room has an even number of doors, including doors that lead outside. Is the total number of outside doors even or odd?

## Forewarned

From *The Booke of Meery Riddles*, 1629:

A soldier that to Black-heath-field went,

Prayed an astronomer of his judgment,

Which wrote these words to him plainly,–

Thou shalt goe thither well and safely

And from thence come home alive againe

Never at that field shalt thou be slaine.

The soldier was slaine there at that field,

And yet the astronomer his promise held.

How?

## Black and White

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?

## Two-Toned

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.

## Moving Up

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?

## Pointing Fingers

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

## Milestones

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?

## 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?

## 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?

## 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?