Professor Stewart's Hoard of Mathematical Treasures (22 page)

Robin Hood and Friar Tuck were engaging in some target practice. The target was a series of concentric rings, lying between successive circles with radii 1, 2, 3, 4, 5. (The innermost circle counts as a ring.)

The target.

Friar Tuck and Robin both fired a number of arrows.

‘Yours are all closer to the centre than mine,’ said Tuck ruefully.

‘That’s why I’m the leader of this outlaw band,’ Robin pointed out.

‘But let’s look on the bright side,’ Tuck replied. ‘The total area of the rings that I hit is the same as the total area of the rings you hit. So that makes us equally accurate, right?’

Naturally, Robin pointed out the fallacy ... but:

Which rings did the two archers hit?

(A ring may be hit more than once, but it only counts once towards the area.)

For a bonus point: what is the smallest number of rings for which this question has two or more different answers?

For a further bonus point: if each archer’s rings are adjacent—no gaps where a ring that has not been hit lies between two that have - what is the smallest number of rings for which this question has two or more different answers?

 

Over the course of one lunar month, the phases of the Moon run from new moon to full moon and back again, passing through various intermediate shapes known as ‘crescent’, ‘first quarter’, ‘waning gibbous’, and the like.

The two ‘quarter’ moons are so named because they occur one-quarter and three-quarters of the way through the lunar month, starting from a new moon. At these times the area of the visible part is half the Moon’s face, not one-quarter. But there are two times during the cycle where a crescent moon occurs, whose visible area is exactly one-quarter of the area of the lunar disc.

When is the area of the crescent one-quarter of the area of the disc?

To simplify the geometry, assume that the Moon is a sphere, and the orbits of both the Moon (round the Earth) and the Earth (round the Sun) are circles lying in the same plane, with both bodies moving at a constant speed. Then the length of a lunar month will also be constant. Assume, too, that the Sun is so far away that its rays are all parallel, and the Moon is sufficiently distant for its image as seen from Earth to be obtained by parallel projection - as if every point on the Moon were transferred to a screen along a line meeting the screen at right angles. (However, you have to replace the real Moon by a much smaller one, otherwise its image ‘in’ the eye would be 3,474 kilometres, or 2,159 miles, across.)

Parallel projection of the Moon’s features on to a screen.

None of these assumptions is true, but they’re good approximations, and the geometry gets a lot harder without them.

 

‘This is a one-line proof - if we start sufficiently far to the left.’

In Mathematical Carnival, the celebrated recreational mathematician Martin Gardner tells us: ‘When a puzzle is found to contain a major flaw - when the answer is wrong, when there is no answer, or when, contrary to claims, there is more than one answer or a better answer - the puzzle is said to be “cooked”.’ Gardner gives several examples, the simplest being a puzzle he had set in a children’s book. In the array of numbers circle six digits to make the total of circled numbers equal 21. See page 304 for Gardner’s answer, why he had to cook the puzzle, how he did that, and how one of his readers cooked his cook. Both solutions are what Gardner calls a quibble-cook, because they exploit an imprecise specification in the question.

Gardner, a puzzle expert, also mentions a more serious example of cookery involving the two arch-rivals of late 19th and early 20th century puzzling, the American Sam Loyd and the Englishman Henry Ernest Dudeney. The problem was to cut a mitre (a square with one triangular quarter missing) into as few
pieces as possible, so that they could be rearranged to make a perfect square. Loyd’s solution was to cut off two small triangles and then use a ‘staircase’ construction - four pieces in all.

After Loyd had published his solution in his Cyclopaedia of Puzzles, Dudeney spotted an error, and found a correct solution with five pieces. The easier question here is: what was the mistake? The harder one is to put it right.