Two runners start from the same point on a circular track and run at different constant speeds. If they run in opposite directions on the track, they meet after a minute. If they run in the same direction, they meet after an hour. What’s the ratio of their speeds?

In an office the boy owed one of the clerks threepence, the clerk owed the cashier twopence, and the cashier owed the boy twopence. One day the boy, having a penny, decided to diminish his debt, and gave the penny to the clerk, who in turn paid half his debt by giving it to the cashier, the latter gave it back to the boy, saying, ‘That makes one penny I owe you now;’ the office boy again passed it to the clerk, who passed it to the cashier, who in turn passed it back to the boy, and the boy discharged his entire debt by handing it over to the clerk, thereby squaring all accounts.

— John Scott, The Puzzle King, 1899

The Royal Statistical Society’s Christmas Quiz runs through January 29 this year — 18 fiendish puzzles to mark the tradition’s 30th anniversary.

“Cracking the puzzles below will require a potent mix of general knowledge, logic, lateral thinking and searching skills — but, as usual, no specialist mathematical knowledge is needed.”

Entry is free and open to everyone. The top two entries will receive £100 and £50 in gift vouchers of their choice, and the quizmaster has pledged a donation of £900 to charities nominated by the top performers.

See the quiz web page for rules, entry instructions, and some tips for budding solvers.

A perplexing question by the Soviet science writer Yakov Perelman:

If a clock takes three seconds to strike three, how long does it take to strike seven?

A problem from the October 1962 issue of Eureka, the journal of the Cambridge University Mathematical Society:

Tom is twice as old as Dick was when Tom was half as old as Dick will be when Tom is twice as old as Dick was when Tom was a year younger than Dick is now. Dick is twice as old as Tom was when Dick was half as old as Tom was when Dick was half as old as Tom was two years ago. How old are Dick and Tom?

A problem by Polish mathematician Paul Vaderlind:

Each child in a school plays either tennis or soccer. One-ninth of the tennis players also play soccer, and one-seventh of the soccer players also play tennis. Do more than half the children play tennis?

In a position puzzle, a phrase is meant to be inferred from the position of words on a page. A familiar example is

stand   take    to     takings.
  I     you    throw      my

This can be read “I understand you undertake to overthrow my undertakings.”

“Sometimes the difficulty is increased by using letters and making them suggest words,” noted Household Words in 1882. It offered this example, adding, “This requires some little thought”:

What does it say?

Here are three cups, one upside down.

Turning over exactly two cups with each move, can you turn all cups right-side-up in no more than six moves?

If it’s possible, show how; if it’s not, say why.

Mathematician Matthew Scroggs has released this year’s Christmas card for Chalkdust magazine.

Solve 10 mathematical puzzles and the answers will guide you in coloring the picture.

You can download the card as a PDF or complete it online.

A puzzle from Henry Dudeney’s Modern Puzzles and How to Solve Them, 1926:

This is a rough sketch of the finish of a race up a staircase in which three men took part. Ackworth, who is leading, went up three risers at a time, as arranged; Barnden, the second man, went four risers at a time, and Croft, who is last, went five at a time.

Undoubtedly Ackworth wins. But the point is, How many risers are there in the stairs, counting the top landing as a riser?

I have only shown the top of the stairs. There may be scores, or hundreds, of risers below the line. It was not necessary to draw them, as I only wanted to show the finish. But it is possible to tell from the evidence the fewest possible risers in that staircase. Can you do it?