## The Lodging-House Difficulty

A puzzle by Henry Dudeney:

The Dobsons secured apartments at Slocomb-on-Sea. There were six rooms on the same floor, all communicating, as shown in the diagram. The rooms they took were numbers 4, 5, and 6, all facing the sea.

But a little difficulty arose. Mr. Dobson insisted that the piano and the bookcase should change rooms. This was wily, for the Dobsons were not musical, but they wanted to prevent any one else playing the instrument.

Now, the rooms were very small and the pieces of furniture indicated were very big, so that no two of these articles could be got into any room at the same time. How was the exchange to be made with the least possible labour? Suppose, for example, you first move the wardrobe into No. 2; then you can move the bookcase to No. 5 and the piano to No. 6, and so on.

It is a fascinating puzzle, but the landlady had reasons for not appreciating it. Try to solve her difficulty in the fewest possible removals with counters on a sheet of paper.

## Seven Tails

Here are seven pennies, all heads up. In a single move you can turn over any four of them. By repeatedly making such moves, can you eventually turn all seven pennies tails up?

## Opposites Exact

Prove that, at any given moment, there are two points on the equator that are diametrically opposed yet have the same temperature.

## Injured List

Another puzzle from Henry Dudeney:

“It is a glorious game!” an enthusiast was heard to exclaim. “At the close of last season, of the footballers of my acquaintance, four had broken their left arm, five had broken their right arm, two had the right arm sound, and three had sound left arms.” Can you discover from that statement what is the smallest number of players that the speaker could be acquainted with?

## The Dwarfs Problem

From the 1977 all-Soviet-Union Mathematical Olympiad:

Seven dwarfs are sitting at a round table. Each has a cup, and some cups contain milk. Each dwarf in turn pours all his milk into the other six cups, dividing it equally among them. After the seventh dwarf has done this, they find that each cup again contains its initial quantity of milk. How much milk does each cup contain, if there were 42 ounces of milk altogether?

## Tiling Task

We’ve removed two squares from this 7×8 grid, so that it numbers 54 squares. Can it be covered orthogonally with tiles like the one at right, each of which covers exactly three squares?

## The Two Poles

A puzzle by Henry Dudeney:

A man planted two poles upright in level ground. One pole was six and a half feet and the other seven feet seven inches above ground. From the top of each pole he tied a string to the bottom of the other — just where it entered the ground. Now, what height above the ground was the point where the two strings crossed one another? The hasty reader will perhaps say, “You have forgotten to tell us how far the poles were apart.” But that point is of no consequence whatever, as it does not affect the answer!

## Out of Rank

A puzzle from John Beasley’s *The Mathematics of Games* (2006): Black has just moved. What is the smallest number of moves that can have been played in this game?

## Drinking Problem

I have a 16-ounce bottle of wine and want to make it last as long as possible, so I establish the following plan: On the first day I’ll drink 1 ounce of wine and refill the bottle with water. On the second day I’ll drink 2 ounces of the mixture and refill the bottle with water. On the third day I’ll drink three ounces of the mixture and again refill the bottle with water. If I continue until the bottle is empty, how many ounces of water will I have drunk?

## Weighty Matters

I’ve accidentally turned the calibration dial on my bathroom scale, so its readings are skewed by a consistent amount. Apart from that it works fine, though. When I stand on the scale it reads 170 pounds, when my wife stands on it it reads 130, and when we stand on it together it reads 292 pounds. How should I adjust the scale?

## Black and White

From Alexander H. Robbins, *A Collection of Chess Problems*, 1887. White to mate in two moves.

## Waste Not …

A puzzle from Polish mathematician Paul Vaderlind:

John is swimming upstream in a river when he loses his goggles. He lets them go and continues upstream for 10 minutes, then decides to turn around and retrieve them. He catches up with them at a point one half mile from the point where he lost them. Is the river flowing faster than 1 mile per hour? (Assume that John swims at the same strength throughout.)

## Twice Blessed

When Linus Pauling won his second Nobel Prize in 1962, he joked that receiving his second Nobel was less remarkable than receiving his first: The chance of anyone receiving his first Nobel Prize is one in several billion (the population of the world), while the chance of receiving his second is one in several hundred (the number of living people who have received one prize).

What’s wrong with this argument?

## Side Issue

A conundrum by Russian puzzle maven Boris Kordemsky:

A work train composed of a locomotive and five cars has just stopped at a railway station when word comes that a passenger train is approaching. The smaller train must make way for it to pass through, but the station has only one siding, and this will accommodate only three cars (or an engine and two cars). How can it arrange to let the passenger train through?

## Flying Fur

Sam Loyd devised this puzzle for P.T. Barnum:

A trained cat and dog run a race, one hundred feet straight away and return. The dog leaps three feet at each bound and the cat but two, but then she makes three leaps to his two. Now, under those circumstances, what are the possible outcomes of the race?

## Lattice Work

Plot five points at random at the intersections of a coordinate grid. Between each pair of points a line segment can be drawn. Prove that the midpoint of at least one of these segments occurs at an intersection of grid lines.

## Black and White

By William Crane Jr., from the Sydney *Town and Country Journal*, 1877. White to mate in two moves.

## High Hopes

A worm crawls along an elastic band that’s 1 meter long. It starts at one end and covers 1 centimeter per minute. Unfortunately, at the end of each minute the band is instantly and uniformly stretched by an additional meter. Heroically, the worm keeps its grip and continues crawling. Will it ever reach the far end?

## Black and White

By C.H. Wheeler, from the *Dubuque Chess Journal*, December 1877. “White to play and compel self-mate in two moves” — that is, White must force Black to checkmate him in two moves.

## Heads and Tails

Let’s play a coin-flipping game. At stake is half the money in my pocket. If the coin comes up heads, you pay me that amount; if it comes up tails, I pay you.

Initially this looks like a bad deal for me. If the coin is fair, then on average we should expect equal numbers of heads and tails, and I’ll lose money steadily. Suppose I start with $100. If we flip heads and then tails, my bankroll will rise to $150 but then drop to $75. If we flip tails and then heads, then it will drop to $50 and then rise to $75. Either way, I’ve lost a quarter of my money after the first two flips.

Strangely, though, the game is fair: In the long run my winnings will exactly offset my losses. How can this be?