Impossible sums

Which numbers cannot be written as the sum of two or more consecutive numbers?
Age
14 to 18
Challenge level
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Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving
Being curious Being resourceful Being resilient Being collaborative

Problem

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?

Proof 1

Proof 2

Proof 3



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.

Proof of Converse

 

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