The evolution of transcendence into a fertile theory with numerous and widespread applications has been an especially exciting development of modern mathematics. The subject was originated by Liouville, Hermite and Lindemann during the last century. Liouville showed that there is a limit to the precision by which an algebraic number, not itself rational, can be approximated by rationals and thereby gave the first examples of transcendental numbers; in fact it suffices to take any non-terminating decimal with sufficiently long blocks of zeros or any continued fraction in which the partial quotients increase sufficiently rapidly.

This article begins with a new look at earlier work on this topic and continues with the presentation of a graphical technique for the determination of the months of a year for which the 13th is a Friday.

It might be thought that the likelihood of the 13 th day of a month being a Friday is the same as that of the 13th being any other day, but this is not so. The full day/date repeat cycle is 400 years, this being the interval between century years which are leap years. The outcome of an extraordinary counting exercise some 50 years ago by a 13-year-old Eton schoolboy, S. R. Baxter [1, 2], was to show that, over this period, the 13th of a month will be a Friday at least once more than any other day. Of necessity, Baxter’s calculation was intricate and an independent confirmation was undertaken as a matter of interest. The 400-year period beginning 1 March 2014 was considered. The work is described in four stages.

1. The dynamics of aeroplane motion is a comparatively new subject, and one that arouses interest in all students of mechanics. Although a full study of aeronautics involves the use of complicated analysis, yet some really useful information about aeroplane motion can be obtained with quite simple mathematics. It is the object of this paper to draw attention to such methods and results. It will be seen that some of the investigations can be understood by mere beginners in mechanics; others are more suitable for such students as have had considerable experience of mechanics.

Here we give an account of two appealingly simple techniques to generate monotonic sequences that were developed by the author and Grahame Bennett. Sadly, Bennett died in 2016 after a lifetime of distinguished contributions in the field of inequalities. This article is dedicated to him.

Our original source for this investigation was an Open University video on Secondary classroom practice (OU video no. PM644). Since then, however, we have seen the problem mentioned in [1], [2] and [3]. The investigation as presented to pupils is as follows:

Choose two integers between 0 and 9 (not both 0) and write them down. As an example, we shall choose 0 and 2. Write down their sum, doing the addition modulo 10. Then write down the sum of the last two numbers in the sequence, again doing the addition modulo 10. Repeat this process. Our example gives the sequence

Two recent problems appearing in die September 1997 and September 1999 issues of the NCTM publication Mathematics Teacher asked the following:

A. An integer is called decreasing if each digit is less than the one to its left. Determine the number of decreasing integers between 2000 and 7000.

B. A 5-digit integer is a peak number if its digits strictly increase to the middle one then strictly descend. Determine the number of 5-digit peak numbers that are greater than 70,000.