Challenge Level

9 can be written as the sum of two consecutive numbers:

9 = 4 + 5

9 can also be written as the sum of three consecutive numbers:

9 = 2 + 3 + 4

**Can you find how to write 10, 11, 12, 13, 14 and 15 as the sum of two or more consecutive numbers?**

10 = 1 + 2 + 3 + 4

11 = 5 + 6

12 = 3 + 4 + 5

13 = 6 + 7

14 = 2 + 3 + 4 + 5

15 = 7 + 8 and 4 + 5 + 6 and 1 + 2 + 3 + 4 + 5

**Convince yourself that it is impossible to write 8 and 16 as the sum of two or more consecutive numbers.**

What other numbers is it impossible to write as the sum of two or more consecutive numbers?

Mathematicians aren't usually satisfied with a few examples to convince themselves that something is always true, and look to proofs to provide rigorous and convincing arguments and justifications.

Can you prove that powers of 2 cannot be written as the sum of two or more consecutive numbers?

Once you have had a think about this, you might like to take a look at these three different proofs which have been scrambled up. Can you rearrange them back into their original order?

If you can find a proof which is different to the ones above, then please do let us know by submitting it as a solution.

So far we have we have asked you to consider how you can prove that powers of $2$ cannot be written as a sum of consecutive numbers. **But is the converse also true, i.e. can any number which is not a power of $2$ be written as a sum of consecutive numbers?** Once you have had a think about this statement, you might like to take a look at the scambled-up proof below.