**For each of the cases below, try some numerical examples to convince yourself that each statement is true.**

**Then try to provide convincing pictorial and/or algebraic arguments that they are always true.**

- Two consecutive numbers add to give an odd number

- The product of two consecutive numbers is even

- The sum of four consecutive numbers is never a multiple of $4$

- Two odd numbers add to give an even number

- The pattern below continues forever: $$7^2=6^2 + 6 + 7 $$ $$8^2 = 7^2 + 7 + 8$$ $$9^2 = 8^2 + 8 + 9$$
- Squaring an odd number always gives an odd number

- If a square number is multiplied by a square number the product is a square number

### Final Challenge

Can you discover any other number rules and provide convincing arguments that they are always true?

*Did you know ... ?*

Although number theory - the study of the natural numbers - does not typically feature in school curricula it plays a leading role in university at first year and beyond. Having a good grasp of the fundamentals of number theory is useful across all disciplines of mathematics. Moreover, problems in number theory are a great leisure past time as many require only minimal knowledge of mathematical
'content'.

*We are very grateful to the Heilbronn Institute for Mathematical Research for their generous support for the development of this resource.*