Suppose a man sets out to climb a mountain at sunrise, arriving at the top at sunset. He sleeps at the top and descends the following day, traveling somewhat more quickly downhill. Prove that there’s a point on the path that he will pass at the same time on both days.

A puzzle from Henry Dudeney’s Amusements in Mathematics (1917):

There is a certain village in Japan, situated in a very low valley, and yet the sun is nearer to the inhabitants every noon, by 3,000 miles and upwards, than when he either rises or sets to these people. In what part of the country is the village situated?

Place two white rooks and a white knight on this board so that Black is checkmated:

The solution need not be reachable in a normal game.

In 1735, an anonymous “lover of mathematicks” offered the following conundrum:

“‘Tis certainly Matter of Fact, that three certain Travellers went on a Journey, in which, tho’ their Heads travelled full twelve Yards more than their Feet, yet they all return’d alive, with their Heads on.”

How is this possible?

Suppose I show you two old coins. One is dated 51 B.C., and the other is marked George I. Which is authentic?

If you were a British soldier in Malta in the 19th century, you might receive this card from a local dram-shop:

What does it mean?

From Amusements in Mathematics by Henry Ernest Dudeney (1917):

The men in the illustration are disputing over the liquid contents of a barrel. What the particular liquid is it is impossible to say, for we are unable to look into the barrel; so we will call it water. One man says that the barrel is more than half full, while the other insists that it is not half full. What is their easiest way of settling the point? It is not necessary to use stick, string, or implement of any kind for measuring. I give this merely as one of the simplest possible examples of the value of ordinary sagacity in the solving of puzzles. What are apparently very difficult problems may frequently be solved in a similarly easy manner if we only use a little common sense.

Okay, I’ll ask three questions, and if you miss one I get your house. Fair enough? Here we go:

1. A clock strikes six in 5 seconds. How long does it take to strike twelve?
2. A bottle and its cork together cost $1.10. The bottle costs a dollar more than the cork. How much does the bottle cost?
3. A train leaves New York for Chicago at 90 mph. At the same time, a bus leaves Chicago for New York at 50 mph. Which is farther from New York when they meet?

Don’t be hasty — your house is on the line.

This question was proposed in the Scientific American, in 1868: ‘How many revolutions upon its own axis, will a wheel make in rolling once around a fixed wheel of the same size?’

The question brought to the editor of that paper many replies all claiming to have solved it. Yet the replies were about equally divided as to the number of revolutions, one part claiming one revolution and the other two revolutions. So much interest was manifested in it that Munn & Co. published The Wheel, June, 1868. It contains 72 pages, giving many of the solutions, illustrated by many diagrams.

— Miscellaneous Notes and Queries, August 1889

So who’s right?