### Pair Sums

Five numbers added together in pairs produce: 0, 2, 4, 4, 6, 8, 9, 11, 13, 15 What are the five numbers?

### Summing Consecutive Numbers

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

### Big Powers

Three people chose this as a favourite problem. It is the sort of problem that needs thinking time - but once the connection is made it gives access to many similar ideas.

# Combining Lengths

##### Stage: 3 and 4

This resource is part of "Freedom and Constraints"

These diagrams show how you can use rods of lengths 3, 4 and 6 units to measure some lengths:

This set of three rods is special because you can measure all the whole number lengths from 1 to 7 units:

The diagrams above show how to make lengths of 1, 3, 5 and 7.
There are rods for 4 and 6. Can you see how to make 2?
Can you convince yourself you cannot make 8?

Can you find another set of three rods that can be used to measure all the lengths from 1 to 7? What about 1 to 8? 1 to 9? 1 to ...?

How far can you go?

What other interesting mathematical questions can you think of to explore next?
We have thought of some possibilities:

Is there only one set of three rods that allows you to measure the largest number of consecutive lengths?

What if you could use a set of four rods?...

Is there a way to decide how long the rods should be for any set of rods?

Is there a quick way to work out how far a set of rods will allow you to measure?

What would happen if you could only add lengths together, and couldn't subtract?

We'd love you to share the questions you've come up with. Tell us also how you got started and any conclusions you have arrived at.
Send us your thoughts; we'll be publishing a selection.