Number theory

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

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: ×2 and -5. What do you think?

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

Just because a problem is impossible doesn't mean it's difficult...

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

Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?

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

Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. Let b(n) be the number of ways of expressing n as an ordered sum of integers greater than 1. (i) Calculate a(n) and b(n) for n<8. What do you notice about these sequences? (ii) Find a relation between a(p) and b(q). (iii) Prove your conjectures.

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