X is a moveable point on the hypotenuse, and P and Q are the feet of the perpendiculars from X to the sides of a right angled triangle. What position of X makes the length of PQ a minimum?

A 10x10x10 cube is made from 27 2x2 cubes with corridors between them. Find the shortest route from one corner to the opposite corner.

The graph represents a salesman’s area of activity with the shops that the salesman must visit each day. What route around the shops has the minimum total distance?

There are 12 identical looking coins, one of which is a fake. The counterfeit coin is of a different weight to the rest. What is the minimum number of weighings needed to locate the fake coin?

It is possible to dissect any square into smaller squares. What is the minimum number of squares a 13 by 13 square can be dissected into?

Three rods of different lengths form three sides of an enclosure with right angles between them. What arrangement maximises the area

A ladder 3m long rests against a wall with one end a short distance from its base. Between the wall and the base of a ladder is a garden storage box 1m tall and 1m high. What is the maximum distance. . . .

Christmas trees are planted in a rectangular array of 10 rows and 12 columns. The farmer chooses the shortest tree in each of the columns... the tallest tree from each of the rows ... Which is. . . .

Which set of numbers that add to 10 have the largest product?

A manager of a forestry company has to decide which trees to plant. What strategy for planting and felling would you recommend to the manager in order to maximise the profit?

Chris is enjoying a swim but needs to get back for lunch. If she can swim at 3 m/s and run at 7m/sec, how far along the bank should she land in order to get back as quickly as possible?