After the publication of the Mathematical Association’s booklet on sixth-form work in pure mathematics in 1967, P. R. Buckland published the results of a short multiple-choice test, given to graduates training to teach mathematics, in the Gazette in 1969. He discovered what he called “alarming gaps” in students’ mathematical backgrounds. Is the situation any different 15 years later? In an attempt to obtain an answer, Buckland’s test (with a few changes and a new ‘historical’ section) was given to a group of graduates and undergraduates.

The genesis of this problem, which was communicated to the first author by Robert Vertes, is in a proposed ‘ice-breaking’ activity at a party attended by an even number of guests. Each guest is assigned a unique number from the set {1, 2, …, n}, where n is even, and their task is to form themselves into ½n pairs so that the sum of the numbers in each pair is a perfect square. It is not difficult to verify that this cannot be accomplished with fewer than eight people, while eight people can be split into the four pairs {1,8}, {2,7}, {3,6} and {4,5}, each summing to the square 9. Sixteen people can also be split into eight pairs with the required property, viz. the four pairs listed above together with {9, 16}, {10,15}, {11, 14} and {12, 13}. There is no requirement that every pair must sum to the same square.

An arithmatical amusement, said to have been first propounded in 1881, is the expression in the expression in the ordinary arithmatic and algebric notation of the consecutive integers from 1 upwards, as far as practicable, by the use of four “ 4 ”. I have mentioned the problem in my Mathematical Recreations, but my friend Mr. Oscar Eckenstein has now carried the solutions considerly further: I think a bare statement of our procedure may be of interest, and perhaps some readers ot the Gazette may amuse themselves by filling in the details ot the analysis or applying similar methods to higher numbers.

A surveyor often encounters problems like Problem 1:

Determine the length of side AC of the right triangle ABC with the right angle at B. A lake intersecting the sides AC and CE prevents the direct measurement of their sides.