Working systematically

Tina has chosen a number and has noticed something about its factors. What number could she have chosen? Are there multiple possibilities?

How many different phone numbers are there starting with a 3 and with at most two different digits?

Five equations and five unknowns. Is there an easy way to find the unknown values?

Investigate the transformations of the plane given by the 2 by 2 matrices with entries taking all combinations of values 0, -1 and +1.

Take three whole numbers. The differences between them give you three new numbers. Find the differences between the new numbers and keep repeating this. What happens?

How would you design the tiering of seats in a stadium so that all spectators have a good view?

How can you quickly sort a suit of cards in order from Ace to King?

Can you work out where these 5 riders came in a not-quite-so-famous bike race?

In 1% of cases, an HIV test gives a positive result for someone who is HIV negative. How likely is it that someone who tests positive has HIV?

DNA profiling is an invaluable tool for the police. However, when it comes to probability, things aren't always as straightforward as they seem.