Being resilient

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

Can you work out the fraction of the original triangle that is covered by the inner triangle?

Can you find the area of a parallelogram defined by two vectors?

Various solids are lowered into a beaker of water. How does the water level rise in each case?

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?

Can you work out the probability of winning the Mathsland National Lottery?

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.