Number theory

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

Can you find ways to put numbers in the overlaps so the rings have equal totals?

Which numbers can we write as a sum of square numbers?

Can you produce convincing arguments that a selection of statements about numbers are true?

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?

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?

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

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?