An annotated translation of Propositiones ad acuendos juvenes, the oldest mathematical problem collection in Latin, attributed to Alcuin of York

In researching the history of recreational mathematics, one finds that the Propositiones ad acuendos juvenes of Alcuin of York is frequently cited as the earliest problem collection in Latin. The text contains 56 problems, including 7 major types of problem which appear for the first time, and 2 major types which appear for me first time in the West. By any standard such a collection is of major historical interest, so it is surprising that a critical edition was not prepared until Folkerts' edition of 1978, and yet more surprising that no English translation has ever been produced.

Several games, such as the “Window Reader”, “Nim”, and “Bachet’s weights problem”, have a theory dependent on the scale of 2. (See, for example, Rouse Ball, Mathematical Recreations and Essays.) Here we give a discussion and some generalisations of two such games, the “Tower of Hanoi” (Rouse Ball, 11th ed., p. 303) and the “Chinese Rings” (Rouse Ball, p. 305).

Tetris is a computer game which has obsessed many computer users and attracted much attention, despite the simplicity of its rules. This paper addresses the question: ‘can you “win” the game Tetris?’

Designed by Soviet mathematician Alexey Pazhitnov in the late eighties and imported to the United States by Spectrum Holobyte, Tetris won a record number of software awards in 1989. Versions of Tetris are sold for most personal computers. There are Tetris arcade games, Tetris Nintendo cartridges, and hand-held Tetris games; Tetris has been played on machines ranging from mainframes to calculators. The game's success has prompted the invention of several similar games, including Hextris, Welltris, and Wordtris.

If we have a system of m linear equations to solve, it is a great simplification to write them in matrix form

Ax = b,

where A is an m x n matrix of coefficients, b is an m-dimensional vector of constants and x is an n-dimensional vector of unknowns.

There is a problem which is much in favour at present of which one version is this : a man has 12 pennies among which there may be a counterfeit coin, which can only be told apart by its weight being different from the others. How can one tell in not more than three weighings whether there is a counterfeit penny, if so which one it is, and whether it is heavier or lighter than a normal penny? (Math. Gaz., XXIX (1945), pp. 227–229; XXX (1946), pp. 231–234.) This puzzle seems to have originated in America. The purpose of the present note is to make clear what is the best possible solution in the most general case of any version of the problem. No originality is claimed for these solutions, as no doubt many of them have been obtained independently many times before : but I am not aware that any complete and systematic account has yet been published.

It is well known that if two people wish to share a cake fairly then one cuts it into two pieces and the other person chooses which of the pieces to have. This article considers the problem of sharing the cake fairly when there are more than two people. We are looking for procedures which ensure justice for an honest person even when there is dishonest collusion by the other people.

The problem is to arrange the numbers 1, 1, 2, 2, …, n, n in a sequence (without gaps) in such a way that for r = 1, 2, …, n the two r’s are separated by exactly r places; for example

Priday has shown in the preceding paper that for every n there exists either such a perfect sequence (as he calls it) or else a hooked sequence, with a gap one place from one end, for example

The differential triangle of Leibniz for a real function f is found by taking an increment dx in the variable x, finding the corresponding increment dy in y = f(x) and drawing the ‘triangle’ in Fig. 1. Here ds is the increment in the length of the graph, where

and the derivative of f is

Let G denote a finite non-commutative group. What can we say about the number of ordered pairs (x,y), where x,y ϵ G, such that xy = yx?

In order to prepare ourselves to answer this question, let us examine the Cayley table of S3, the symmetric group on three symbols. We take as generators the reflection (12) and the rotation (123). It is clear that we are examining the Cayley table for symmetry about its main diagonal.

The Fibonacci sequence

0, 1, 1, 2, 3, 5, 8, 13, …

is defined by u0 = 0, u1 = 1, and un = un−1 + un−2 for . There are many elementary formulae relating the various un, most of which, since the sequence is defined inductively, are themselves usually proved by induction.

The process of randomisation has in recent years come to play such a central part in experimental design that it is of some interest to find that it affords a means of resolving one of the oldest paradoxes which arose in discussions of gaming.

Readers of Todhunter’s The Mathematical Theory of Probability will recall his account (sections 187-190, pp. 106-110) of the correspondence between Montmort and Nicolas Bernoulli on the rule by which the players might guide themselves most advantageously in the game called “le Her”.