This article begins with the consideration of a short program designed to produce random values from a normal distribution and goes on to suggest how a similar program might be used to introduce confidence intervals to a group of A-level students. The programs are written in BASIC for the BBC microcomputer.

The fact that a mechanical system so well known to so many in its practical aspect as the swing has been thoroughly neglected in textbooks and problem papers may justify a few remarks on it here.

The only reference to the swing which I have been able to find is the question requiring it to be shown that the amplitude of the swing may be increased by internal action only, if, when the swing passes through its lowest position, the distance of the centre of gravity from the point of suspension is (suddenly) shortened from l to (l – h), the full distance l being restored at the end of the swing.

No field of endeavour, trivial or non-trivial, escapes man’s quest for the superlatives—always the smallest or the largest, the fewest or the most is his goal. Whether this drive stems from a thirst for knowledge or a desire for leisure in which to pursue other work is a good question. In this massive and perpetual search for the ultimate a trivial game is no exception. No doubt a fantastic amount of time has been spent seeking to determine how many little counters could be arranged in a certain pattern, then this pattern re-arranged by jumping one unit over the other, removing the counter jumped, until the greatest number is lined up to be taken in a single move, or the pattern is reduced to a single unit with the fewest moves. Erom available published accounts, the single move limit has been 8 and the fewest moves have been 16 since this particular pattern called a 33-Solitaire board was conceived, the 33 referring to the number of spaces in this pattern.

In his recent article [1] F. J. Budden produced a binary operation on the set R0 of non-zero real numbers that was associative but not commutative, namely

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.

The method usually employed for the extraction of square roots involves a series of successive approximations, one for each significant decimal figure of the root. A binomial series may be found which converges quicker than Σ10-n, but this involves a number of separate divisions. Iteration by dividing an approximate value of the root into its square and taking the mean nearly doubles the number of significant figures at each operation. The methods here given converge rather faster than this type of iteration, and enables a root to be expressed as a simple infinite series. They arose from a question put to me by Mr. K. A. Kermack, as to how an ancient Roman could have obtained a good approximation to a square root as a rational fraction. I believe them to be novel, at least in part. It will clearly be sufficient to give methods for finding the square root of a positive integer k.