Mathematicians are always looking for efficient methods for solving problems. How efficient can you be?

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number?

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears.

Square numbers can be represented on the seven-clock (representing these numbers modulo 7). This works like the days of the week.

A Latin square of order n is an array of n symbols in which each symbol occurs exactly once in each row and exactly once in each column.

Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?

A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?

Here are many ideas for you to investigate - all linked with the number 2000.

Time for a little mathemagic! Choose any five cards from a pack and show four of them to your partner. How can they work out the fifth?