What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?

Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

Andy is desperate to reach John o'Groats first. Can you devise a winning race plan?

Four friends must cross a bridge. How can they all cross it in just 17 minutes?

The net of a cube is to be cut from a sheet of card 100 cm square. What is the maximum volume cube that can be made from a single piece of card?

What is the longest stick that can be carried horizontally along a narrow corridor and around a right-angled bend?

Given the graph of a supply network and the maximum capacity for flow in each section find the maximum flow across the network.

How many tours visit each vertex of a cube once and only once? How many return to the starting point?

A box of size a cm by b cm by c cm is to be wrapped with a square piece of wrapping paper. Without cutting the paper what is the smallest square this can be?