### Fixing the Odds

You have two bags, four red balls and four white balls. You must put all the balls in the bags although you are allowed to have one bag empty. How should you distribute the balls between the two bags so as to make the probability of choosing a red ball as small as possible and what will the probability be in that case?

### Scratch Cards

To win on a scratch card you have to uncover three numbers that add up to more than fifteen. What is the probability of winning a prize?

### Concrete Calculation

The builders have dug a hole in the ground to be filled with concrete for the foundations of our garage. How many cubic metres of ready-mix concrete should the builders order to fill this hole to make the concrete raft for the foundations?

# Escalator

##### Age 14 to 16 Challenge Level:

Will of The Ridgeway School; Xing Cong of Raffles Institution, Singapore; Elizabeth of Ipswich High School; and Alexandra of Bancroft's School, Woodford Green, Essex; all used very similar methods. This is Alexandra's beautifully clear solution:

There are 100 steps. When the taller man reaches the top the shorter man will have climbed one third of 75 steps i.e. 25 steps. This means that the shorter man is half way up the escalator when the taller man has reached the top. At this moment, there are 50 steps between the men (the difference between 75 and 25) which means that half the escalator is equal to 50 steps. The whole escalator is therefore double this, that is 100 steps.

Although it takes longer and more work, this problem can be solved by supposing there are k steps on the escalator, that the speed of the escalator is v steps per minute and the speeds of the two men are u and 3u steps per minute. Now you need four equations to find k. You can write down these four equations from the distance, speed and time for the smaller man relative to the ground, for the smaller man relative to the escalator, for the taller man relative to the ground and for the taller man relative to the escalator.