Working systematically

Weekly Problem 26 - 2008
If $n$ is a positive integer, how many different values for the remainder are obtained when $n^2$ is divided by $n+4$?

You have twelve weights, one of which is different from the rest. Using just 3 weighings, can you identify which weight is the odd one out, and whether it is heavier or lighter than the rest?

This train line has two tracks which cross at different points. Can you find all the routes that end at Cheston?

Imagine a stack of numbered cards with one on top. Discard the top, put the next card to the bottom and repeat continuously. Can you predict the last card?

This grid can be filled so that each of the numbers 1, 2, 3, 4, 5 appears just once in each row, column and diagonal. Which number goes in the centre square?

How many 10-digit numbers containing only 1s, 2s and 3s can you write?

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

Weekly Problem 11 - 2011
Kanga hops ten times in one of four directions. At how many different points can he end up?

Can you crack these very difficult challenge ciphers? How might you systematise the cracking of unknown ciphers?

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