In the following note, Stirling’s formula for a factorial is first found by a very simple process. The expression so obtained has a meaning for non-integral as well as integral values of the variable. The function it defines is easily seen (§§ 3 ff.) to possess the fundamental properties of the Gamma function, and provides an easy and natural avenue of approach to its study.

The standard tape–recorder consists outwardly of two spools on which the magnetic tape is wound, from the left to the right, between least and greatest radii of a and b. On my machine, a is about 1.05 in. and b is about 2.75 in. Apart from what is on the tape, the only indication at any intermediate stage as to how much playing time is still left is provided by a revolution counter geared to the left hand spool, the one which starts full and finishes empty. The numbers registered are about half the actual number of revolutions made on the left: this arrangement allows some 1400 revolutions to be registered by only three digits. It can be assumed that the tape is fed out at a constant length per unit of time: so the time still to run is proportional to the length still to run. If the total number of revolutions to run is N, how much of the tape ha3 been run off when the reading is n?

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.

In the extension from the use of natural numbers with zero to integers, i.e. the introduction of directed numbers including negative numbers, much experimental work is now being done with the use of mathematical structure. Some attempts are being made to give this as the initial treatment at about the age of 12, and some teachers are getting their children very interested and consider it a correct method. For other children the formal treatment will come later, when they are more mature and begin to query the rules they are using. This may be when they are starting to work ahead of their O level syllabus, the beginning of a more advanced course with work on groups. It could give work for those doing some mathematics when they are beginning to specialise in some other subjects and want to try to understand more about the mathematics they have learnt.

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.