Reverses

A Hungarian problem shortlisted for the 30th International Mathematical Olympiad, 1989:

Around a circular race track are n race cars, each at a different location. At a signal, each car chooses a direction and begins to drive at a constant speed that will take it around the course in 1 hour. When two cars meet, both reverse direction without loss of speed. Show that at some future moment all the cars will be at their original positions.

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Imagine that each car carries a flag, and that when two cars meet they exchange flags. Now each flag travels around the course with a constant direction and speed, and so after 1 hour each flag has returned to its starting position. Because each car reverses direction when it meets its neighbor, the order in which the cars are arranged around the track does not change. This means that the starting position has now been rotated, perhaps onto itself, and that after an integral number of hours the original position must be reached again. See Measured Steps.

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June 2, 2013June 7, 2013 | Puzzles

Black and White

markx chess problem

By B.J.M. Markx, 1897. White to mate in two moves.

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1. Bd3! blocks Black’s d-pawn, forcing him to move a knight or the c-pawn — and accept a fatal check from the queen.

May 31, 2013June 7, 2013 | Puzzles

Weather Report

A puzzle contributed by Howard C. Saar to Recreational Mathematics Magazine, October 1962:

On the day before yesterday, the weatherman said, “Today’s weather is different from yesterday’s. If the weather is the same tomorrow as it was yesterday, the day after tomorrow will have the same weather as the day before yesterday. But if the weather is the same tomorrow as it is today, the day after tomorrow will have the same weather as yesterday.”

It is raining today, and it rained on the day before yesterday. What was the weather like yesterday? (Note: The prediction was correct!)

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Yesterday was fair — no rain.

May 30, 2013June 7, 2013 | Puzzles

Coverup

coverup puzzle diagram

Two squares have been removed from this 8×7 rectangle. Can the remaining 54 squares be tiled orthogonally with 18 3×1 tiles?

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No. Color the squares as shown. Now each tile must cover a blue, a yellow, and a red square. If we’re to cover the figure completely, then it must contain an equal number of squares of each color. But it contains 17 blue squares, 18 yellow squares, and 19 red squares. So the task is impossible.

May 25, 2013May 31, 2013 | Puzzles

Black and White

mccombe chess problem

By Alexander George McCombe. White to mate in two moves.

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1. Qg2! fxg2 2. f4# From the Illustrated London News, 1849.

May 24, 2013May 31, 2013 | Puzzles

Blocked

You have n cubical building blocks. You try to arrange them into the largest possible solid cube, but you find that don’t have quite enough blocks: One side of the large cube has exactly one row too few.

Prove that n is divisible by 6.

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Let x be the number of blocks in a full row. Then n = x³ – x = x(x + 1)(x – 1). These factors are three consecutive integers, which means that at least one of them is even and one is divisible by 3. Since its factors include numbers that are divisible by both 2 and 3, n is divisible by 6. (Thanks, Allan.)

May 20, 2013May 31, 2013 | Puzzles

Black and White

charlick chess problem

By Arthur Charlick. White to mate in two moves.

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1. Ra2! and the knight will mate. From the Australian Chess Annual, 1896.

May 17, 2013May 24, 2013 | Puzzles

Going Up

A problem from the 2003 Moscow Mathematical Olympiad:

A store has three floors, which are connected only by an elevator. At night the store is empty, and during the workday:

(1) Of the customers who enter the elevator on the second floor, half go to the first floor and half to the third floor.

(2) The number of customers who get out the elevator on the third floor is less than 1/3 the total number of customers who get out of the elevator.

Which is greater, the number of customers who go from the first floor to the second on a given workday, or the number who go from the first floor to the third?

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Denote the number of trips from the first floor to the second floor by n₁₂, and so on. Now we can divide all the day’s elevator trips into three groups: trips to the third floor (n₁₃ + n₂₃), trips from the third floor (n₃₁ + n₃₂), and trips between the first and second floors (n₁₂ + n₂₁). By assumption, the first group makes up less than 1/3 of all trips. Now, if the store is empty at night, and if the floors communicate only by elevator, then the total number of customers who enter the third floor on a given day is equal to the number who leave it. This means that the first group above is the same size as the second. And it follows that the last group is the largest: (n₃₁ + n₃₂) = (n₁₃ + n₂₃) < (n₁₂ + n₂₁) If we substitute n₂₁ = n₂₃, we get n₁₃ < n₁₂: More customers go from the first floor to the second than from the first to the third.

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May 16, 2013May 24, 2013 | Puzzles

Too Late?

Paul R. McClenon of Washington D.C. contributed this problem to the January-February 1964 issue of Recreational Mathematics Magazine:

The poor patient read the prescription which would save his life. ‘Mix carefully a one-pint drink, made of scotch whisky and water, mixed one to five (1/6 scotch, 5/6 water). Drink it quickly and go to bed.’

However, the patient finds only the following items at hand:

A one-quart bottle, about half full of scotch whisky.
An eight-ounce glass.
An unlimited supply of water from his faucet.
A sink with a drain.
No other containers or measuring devices.

He can pour from either container to the other, without spilling a drop, and can fill either to the brim without loss. How should he mix the prescription? Will he figure it out in time? Will he be saved? Did a doctor or bartender write this prescription?

The magazine went out of business before it could publish the solution. I’ll leave it to you.

05/17/2013 UPDATE: There seem to be a number of ways to accomplish this. Here’s one:

We need a 16-ounce dose that’s 1/6 whiskey, so the final mixture must contain 2.666 ounces of whiskey.

Fill the 8-ounce glass with whiskey, then empty the jug.
Return the glassful of whiskey to the empty jug and add two glassfuls of water.
Fill the glass with this 2-to-1 mixture of water and whiskey. The 8-ounce glass now contains 2.666 ounces of whiskey, our target.
Empty the bottle, pour the glass’ contents into it, and add one 8-ounce glassful of water.

That leaves us with 16 ounces in the jug, 1/6 of which is whiskey and the rest water, as directed.

Thanks to everyone who wrote in.

May 15, 2013May 17, 2013 | Puzzles