A ferry starting on one side of a river is to cross to the adjacent point on the opposite bank travelling at a constant speed f but moving against a variable river sneed We will ignore starting and stonninc difficulties which only operate for a short interval of time. The ferry moves so that it is always aiming at the target position T. The river is assumed to be straight and the speed of the river is given by the function r(x) which is non-negative, continuous and differentiable for 0 ≤ x ≤ a, where a is the width of the river. We shall show that the solution reduces to two integrations and the condition which determines whetIer the derry can reach its destination will be given.

The most important properties of any distribution are its moments. The first four moments, for example, can be used describe any data set fairly well. Moments can also be used for estimation.

Many readers will be familiar with the sequence of Catalan numbers {Cn: n ≥ 0} and the formula

with its alternative form

These can be proved by using recurrence relations, generating functions or André's reflection principle. A good reference for all of these methods is Martin Griffiths' book [1].

However, none of these approaches strikes me as being naturally combinatorial. A formula such as (1) is often derived by making a list of all the ways of doing something, and then subdividing this list into classes of equal size, so that either one class consists entirely of ‘valid’ cases or there is exactly one ‘valid’ case in each list.

The so-called Archimedean iterative algorithm for calculating π uses a method involving the two equations

(1)

and

(2)

(Note that (1) is the harmonic mean.) Imagine two regular polygons each with the same number of sides, circumscribed and inscribed to a circle of diameter one. The larger one has perimeter a0, the smaller has perimeter b0. (Archimedes used hexagons with and b0 = 3, but regular polygons of any number of sides can begin the iterations.) Since the perimeter of the circle is π we have b0 < π < a0. Now consider regular polygons with twice the number of sides that circumscribe and inscribe the circle and call their perimeters a1 and b1 respectively. These can be calculated, from (2) and (3). (See [1] for a derivation or many explanations on the web.) Continuing in this way we generate the perimeters of inscribed and circumscribed regular polygons, and in each case the number of sides is twice the sides of the previous polygon.

We consider a simple extension of the familiar Tower of Hanoi puzzle. There are three vertical pegs lined up in a row and n discs. The discs have holes in their centres so that they can be threaded on to the pegs. Initially, all n discs are placed on the left-hand peg in non-increasing order of radius to form a tower, as in Figure 1. The object of the game is to transfer the entire tower to the right-hand peg by moving discs from peg to peg, one at a time according to the rules:

• (1) only a disc at the top of a tower may be moved;

• (2) a disc must never be placed on top of a smaller disc.