A topic of current interest is that of finding defining sets (preferably smallest defining sets) for various combinatorial structures such as latin squares and block designs, so it seemed interesting (though of no practical use) to try to solve the same problem for magic squares of small size. (The germ of this idea was suggested in the Gazette Note [1] of 1988 under the sub-heading ‘Squares from clues’.)

Consider the functional equation

Now let and These are elements of a group, G1 say, which is isomorphic to C2, the cyclic group of order 2, as can be seen from the group table:

The beautiful infinite product of radicals

due to Vieta [1] in 1592, is one of the oldest non-iterative analytical expressions for π, In a previous paper [2] the author proved the following two Vieta-like products:

for N even, and

for N odd. Here N is a positive integer, FN and LN are the Fibonacci and Lucas numbers, and is the golden section. (The Fibonacci numbers are F1 = 1, F2 = 1, with the recursion relation , while the Lucas numbers are L1 = 1, L2 = 3 with the same recursion relation )

Arithmetical properties of numbers have been a source of fascination for mathematics students and professionals throughout the ages. In this paper we consider a puzzle of sorts in which we move the first digit of a positive integer A to the last position and establish conditions under which A will be equal to a multiple of the new integer. For instance, if A equals 820512 then A = 4A’where A’ = 205128.

In exploring the probabilities involved in a roll-a-penny game, some properties turned out to be a little more complicated than expected. The main aim here is to find the probability that the player will lose after a certain number of rolls.

Suppose we have a board upon which squares are drawn, and that the side of each square is four times the radius of a penny. (In practice, this means that the squares are of side 4 cm.) The first question is, “What is the probability that the penny lands entirely within a square, assuming that the dividing lines are of negligible thickness?”

Mr. I. R. Vesselo: The problem of selecting a suitable programme of films and film strips for this meeting was not a very difficult one; the choice is very limited. Like the evening-class system, which grew out of a desire, as much to make use of empty classrooms as to educate, the educational film grew partly out of a desire to make wider use of films whose first function was to fill out the normal entertainment programme. An audience would tolerate a little education whilst waiting for the “big picture”

“First books” on mathematics usually begin with numbers. Some histories of mathematics say in so many words. Mathematics begins with numeration. I believe this is true, literally, if mathematics is considered as a set of signs, with rules and relations between them. That is to say: the history of mathematical notation does begin with the numerals. The earliest known mathematical signs are no doubt number-signs, and history of mathematical notation has in fact little else to discuss but numerals up to the 14th century of our era. Of present-day mathematical signs, + and - appeared first in the 15th century, = in the 16th, <and> in the 17th.