The first author discovered the property below while playing with his pocket calculator, and the second author then tried to find out what was behind this property. We enjoyed this voyage of discovery and thought that you might like to share it with us.

Actually, a calculator is not essential, but you will need a square array like in front of you. If your calculator looks different, that is no problem: all standard keyboards will work just as well.

There are n 1 places in the world which I should like to visit. If I take the shortest possible route to visit all of these places, starting and finishing at home, what is the greatest possible distance I need to travel?

In this article we are going to look at the above problem, namely that of finding the maximin circuit length for n points (n 1 and home).

For many sixth-formers and undergraduates, the method of proof-by-induction seems to resemble nothing so much as a confidence trick. They appear to have little or no appreciation of the logical foundation of the method; frequently, they have been given a mysterious recipe which might just as easily be “eye of newt and toe of frog”. They slavishly follow the rules given and so it is little wonder that they are confused when, having been cajoled incessantly not to beg the question, they are now seemingly allowed to assume the answer. However, it is not my intention in this article to present yet another justification for the principle of mathematical induction, important as that might be; in any case, perfectly good articles on the subject have already appeared in the Gazette [1,2].

This article provides solutions to three problems. The solutions are interesting in themselves; but the problems, which are very different in nature, have a common characteristic. Each can be solved relatively quickly—some would say, elegantly—by a leap of intuition. On the other hand, all three can be tackled in a more exploratory, pedestrian fashion. Some people may want to argue that one approach is “better” than another; but a more useful discussion concerns the appropriateness of a method in particular circumstances, whether it provides insight into the underlying structure and principles involved, and indeed whether this is important.

The announced surprise test has long been a source of puzzlement among laymen and mathematicians alike. On the one hand, it is asserted “There will be exactly one test next week” and, on the other, it is asserted “The day of the test will be a surprise”. Simple logical analysis suggests that these two statements are in contradiction, yet, of course, when the test comes the class is surprised. By 1977, Martin Gardner had published a summary of solutions to the paradox gleaned from over 20 articles in the mathematical and philosophical literature. Most of the views expounded by Gardner admit the contradiction and implicitly invoke the logical rule that asserts that anything whatever can follow from a contradiction. This is also the approach of one of Gardner’s readers who purports to prove that the egg (the paradox of the unexpected egg, hanging, test, etc are all isomorphic) must be in box 5, yet his proof works as well for any other box! Others assume that perhaps the teacher (judge, friend with the eggs) may not be telling the truth, whereas one respondent understands "expected" in the statistical sense of “expected value”.

The possible dangers and anomalies arising from the casual interpretation of “mean values” or “averages” are well known. The “velocity problem” is often cited as an example (see e.g. [1,2]). A typical version of this problem is as follows: A cyclist goes from A to B at 20 mph and returns from B back to A at 10 mph. What is the average speed? The casual answer is 15 mph, the arithmetic mean of the two speeds; this is wrong. The correct answer, obtained by dividing the total distance travelled by the total time taken, is obtained as 13.3 mph, and this latter figure is readily shown to be the harmonic mean of the two speeds. Langley goes on to say that “the harmonic mean is often appropriate when the data consists of ratios (miles per hour, . . .)”; this is true but not specifically informative. Reichmann comments that “the arithmetic mean cannot properly be used to average rates of speed” and “the harmonic mean is always the right one to use to average rates . .-.”; these generalisations are incorrect.