PRIZEGIVING PROBLEMS, JULY 2009

Solutions

If you had a tube of Smarties, what’s the minimum number you would have to eat in order to guarantee that you have eaten two of the same colour?

If there are n colours, then n + 1 must be eaten in order to guarantee that you have had two of the same colour (since all n + 1 cannot be different, at least two must be the same).

The value of n has varied over time, but can be estimated by purchasing a tube.  Note that, for any given number of tubes (no matter how large), there is a small (but finite) chance that at least one colour will be missing.

Anne, Beth and Catherine are sitting round a table eating liquorice sweets which make some people’s teeth go black.  Anne is laughing at Beth and Catherine who both have black teeth and they are both laughing as well.  Does Anne have black teeth?

This is a variation of a well known problem.  Anne’s teeth are black.

If Anne’s teeth were not black, Beth would wonder why Catherine was laughing and would realise that her own teeth were black.  People only laugh if they are not laughable themselves, so Beth would no longer be laughing.

What’s the chance that the last Smartie in the tube is blue?

The chance of the last one being blue is the same as the chance of the first one being blue.

Assuming that there are 7 colours, that an equal number of each are manufactured, and that tubes are filled at random, then the answer would be 1/7 (1 in 7).  However, reliable sources indicate that only approximately 12% of Smarties are blue, so the chance is a little less than 1 in 8.

This is all based on the (false) premise that people choose Smarties at random.  If blue Smarties are well liked, then the chance of the last one being blue will be less than the given figure.  If, on the other hand, people are put off by the blue colour, then the chance of the last one being blue may be significantly greater than the estimate given.