### Latin Numbers

Can you create a Latin Square from multiples of a six digit number?

### Summing Consecutive Numbers

15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?

### Number Sandwiches

Can you arrange the digits 1, 1, 2, 2, 3 and 3 to make a Number Sandwich?

# What Does it All Add up To?

##### Age 11 to 18Challenge Level

$5, 6, 7, 8$ are four consecutive numbers. They add up to $26$.

Take other sets of four consecutive numbers and find their total.
Do the totals have anything in common?

Can you find four consecutive numbers that add to $80$?
If not, might it be impossible?
What other even numbers cannot be written as the sum of four 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 the sum of four consecutive numbers is always an even number which is not a multiple of $4$?

Below is a proof that has been scrambled up.
Can you rearrange it into its original order?

Click on student solutions to see some different proofs that students submitted.

Extension:

Can you prove that the sum of five consecutive numbers is always a multiple of $5$?
Can you prove that the sum of six consecutive numbers is always a multiple of $3$ which is not a multiple of $6$ (i.e. an odd multiple of $3$)?

Challenging Extension:

Can you prove the following statements?

• If $n$ is odd, then the sum of $n$ consecutive numbers is always a multiple of $n$.
• If $n$ is even, then the sum of $n$ consecutive numbers is always a multiple of $\dfrac{n} {2}$, but is not a multiple of $n$.

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