One approach to investigating the structure of a group is to examine the family of subgroups of the group. In particular, one often looks for normal subgroups and considers things such as direct products and composition series. However, fruitful investigation usually involves establishing that certain subgroups do, in fact, exist. Hence it is important to find conditions which will guarantee the existence of subgroups. This note indicates the usefulness of one such set of conditions—the Sylow theorems for finite groups.

Although it is difficult to think of a rational reason for being more interested in, say, the square number 698896 than in the square 700569, or for that matter in 4 or 49, there is apparently some psychological reason, and the pursuit of square numbers of particular digital forms has a long history. The publication of the attractive collection of digital curiosities in Note No. 1823 has suggested to me that there may be some interest in further figures from discoveries I have made in analogous fields. References in a few cases indicate previous publication in scattered problem columns.

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

If we form the sum S(n) of the digits of a number n, and then the sum of the digits of S(n), and so on, in a finite number of steps we reach a number of one digit, R(n), which we call the reduce of n.

Starting with a number prime to 9, say 2, consider the reduces of powers of 2; we obtain the sequence 2, 4, 8, 7, 5, 1, 2, 4, 8, ..., which recurs after the sixth term. Consider next a five-digit number, say 16427; from this we derive 15794 formed by replacing each digit of 16427 by the reduce of the sum of the remaining digits. Repeating the transformation, we obtain in turn 73184, 72461, 49751, 48137, 16427, the number being restored to its original value after only six transformations. It will be found that a three-digit number repeats after eighteen transformations and yet for a twenty-digit number only six transformations are needed.

Two players A and B have a game of squash. Player A serves at the start of the game. They play a sequence of rallies. If the player serving the rally wins the rally, he scores a point, and serves the next rally. If he loses the rally, then the other player serves the next rally, with the score unaltered.

Generally the first player to reach nine points wins the game. However, if the score becomes eight-all then the first player to reach eight has a choice: he specifies whether the game is to be played to nine points or to ten ponts.

What is the correct choice? We propose to analyze the problem by use of an elementary probabilistic model.