There is a popular view among sixth form mathematicians, and therefore presumably among some who teach them, that evaluation of an integral by the approximate Trapezium Rule is made more accurate by increasing the value of n.

When I was younger, and had nothing better to do with my spare time, I sometimes used to play my own game of ‘patience’. I would take a pack of 52 playing cards and arrange them in order, ace to king of each suit and then the four suits one on top of the other. I would begin my game by taking the card at the top of the pack and putting it at the bottom; then taking the second card and placing it face upwards on the table; then the third card was transferred to the bottom of the pack; then the fourth card was placed on top of the second on the table; and so on, until all the cards had been placed in a new pile on the table. I would then turn the pack on the table face downwards and perform the whole operation again with the cards arranged in this new order. My aim was to see if, by repeating the whole operation enough times, I could get the cards back into their original order. If I had known that I would need to perform the operation 840 times, I probably would not have bothered to continue with my game.

The June 1975 Gazette (59, No. 408, 98-106) carried an article by Colin Tripp about the so-called “adventitious angles” problem: in the isosceles triangle in Fig. 1, for certain given values of the angles marked a, b, c, to find θ. Correspondence to which this gave rise was summarised in a “progress report” in March 1977 (61, No. 415, 55-58). This cleared up certain points raised by Tripp, using trigonometric methods; and its publication also had the desired effect of stimulating some further response, which has settled most of the outstanding questions. It seems now to be an appropriate time for the editor to produce a “final summary” for those readers who have taken an interest in the problem.

A quadrangle has four vertices, of which no three are collinear, and six sides joining the vertices in pairs. If the angle between each pair of the six sides is an integral multiple of π/n radians, n being an integer, the quadrangle is said to be n-adventitious [1]. A quadrangle is adventitious if it is n- adventitious for some n. For example, the quadrangle BCDE in Fig. 1 (the original adventitious quadrangle from which all the discussion started in [1]) is 18-adventitious. Various problems are posed in [1]; in a suitably generalised form these problems can be summarised as: find all adventitious quadrangles and prove their existence by elementary geometry.

Why is such a nice result rarely, if ever, found in texts? Perhaps because it is usually necessary to use trigonometry to calculate the areas of regular polygons. A regular n-gon inscribed in a unit circle has an area n times that of the isosceles triangle having an edge as base and the centre as apex; that is,

area of regular n-gon = n × (1/2. 1.1. sin (360/n)°) = 1/2n sin (360/n)°.