McMahon's Squares // 183

Explain the mistake to Henry. Then find all such sums, with single non-zero digits in the first two fractions, that are correct.

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Answer on page 291

How Deep is the Well? In one episode of the television series Time Team, the indefatig- able archaeologists want to measure the depth of a mediaeval well. They drop something into it and time its fall, which takes an amazingly long six seconds. You hear it clattering its way down for ages. They come dangerously close to calculating the depth using Newton's laws of motion, but cop out at the last moment and use three very long tape measures joined together instead.

The formula they very nearly state is

s ¼ 1 gt2

2

where s is the distance travelled under gravity, falling from rest, and g is the acceleration due to gravity. It applies when air resistance can be ignored. This formula was discovered experi- mentally by Galileo Galilei and later generalised by Isaac Newton to describe motion under the influence of any force.

Taking g ¼ 10 m sĄ2 (metres per second per second), how deep is the well?

You've got three days to do it.

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Answer on page 292

McMahon's Squares This puzzle was invented by the combinatorialist* P.A. McMahon in 1921. He was thinking about a square that has been divided into four triangular regions by diagonals. He wondered how many different ways there are to colour the various regions,

* A combinatorialist is someone who invents this kind of thing.

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