One question left up in the air from previous weeks was the one about the fly on the lake of jam. We had to show that: No matter where a tiny fly is on the surface of the jam lake shown there is always a pair of points on its edge such that she is midway between them.
Choose a position in which the two distances are clearly unequal (if that's not possible then the proof is already completed) (Fig a). Now imagine pivoting the line about the fly's position. The lengths of XA and XB will change continuously as the line is rotated. Half-way round, they have swapped roles and XA is shorter than XB. Since they start off with XA bigger than XB, and end up with XA shorter than XB, there must be some intermediate position at which the changeover took place: a position where they were equal.
This can be illustrated very neatly by a sketch graph of distance against angle of rotation (Fig b). The detailed shape of the graph doesn't matter much. The key point is that if on the left we depict A as further than B is from X, and vice versa on the right, there must be a crossover somewhere in between. Several of you (eg Bob Parslow of Middlesex) noted that the jam slick given was not just any old jam slick. It had no concavities: it was "everywhere convex".
Points to ponder
1 A mountaineer climbs Mount Neverest. She sets out from base camp at sea-level at noon and reaches the summit at midnight. The next day she starts her descent at noon, reaching base camp at midnight. Was there a point in her journey at which she was at the same height above sea-level at the same time on the two days?
2 Find a single-word anagram of MOUNTAINEER.
3 An equilateral triangle may be cut into a number of equilateral triangles (not necessarily equal). Four is easy and so is, for example, seven (see Fig c). Into which numbers of equilateral triangles is it not possible to cut an equilateral triangle? Prove that any other number is possible!