22. Strictly speaking, this puzzle is impossible to do in two dimensions – in other words, on a flat piece of paper or a screen. Except for this cunning solution: let one of the lines pass THROUGH a house to get to the other houses! Look at the yellow line in this diagram:

In three dimensions, the puzzle is perfectly possible: you can dig under the house and bring lines up from underneath, or you could put lines on poles and bring them in from the air.

And there’s another, advanced solution, using a mathematical shape called a torus (really just a fancy word for “doughnut”): if you drew the houses, utilities and all the lines connecting them on the surface of the torus, with one line going through the doughnut-hole, it would be possible to do it all on one “plane” without crossing or touching. (You could emulate this effect by punching a hole in your paper diagram and drawing on both sides.)

As with so many classic brainteasers, this one is very old – predating electricity or telephones in homes – and actually involves complex mathematics: in this case, a branch of maths called graph theory.

23. A million billion

There are some great online visualisations of very large numbers.

This website (with a rather provocative name) has really excellent graphics illustrating large amounts of money: usdebtkleptocracy

A sombre, but very striking illustration of deaths in Cambodia due to the Khmer Rouge.

A clear, educational site showing large numbers of coins. (And some cute cows.)

In future Mind Games, we’ll look at ways of comprehending other vast numbers, like interplanetary distances and evolutionary time.

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