Being resilient

Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?

Is it possible to find the angles in this rather special isosceles triangle?

What is the largest number which, when divided into these five numbers in turn, leaves the same remainder each time?

A mother wants to share some money by giving each child in turn a lump sum plus a fraction of the remainder. How can she do this to share the money out equally?

Chris and Jo put two red and four blue ribbons in a box. They each pick a ribbon from the box without looking. Jo wins if the two ribbons are the same colour. Is the game fair?

Two boats travel up and down a lake. Can you picture where they will cross if you know how fast each boat is travelling?

Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?

Which numbers can we write as a sum of square numbers?

Six samples were taken from two distributions but they got muddled up. Can you work out which list is which?

In this follow-up to the problem Odds and Evens, we invite you to analyse a probability situation in order to find the general solution for a fair game.