# Sums, Squares and Substantiation

Creating convincing arguments or "proofs", to show that statements are always true, is a key mathematical skill. The problems in this feature offer you the chance to explore number patterns and create proofs to show that these are always true.

Many of the problems in this feature include proof sorting activities which challenge you to rearrange statements in order to recreate clear, rigorous proofs.  There are also a selection of "dodgy proofs" where your challenge is to find out where the logic breaks down.

The live problems will be open for solutions until Monday 31 January.

Plus magazine has a selection of interesting articles about proofs here.

### What Does it All Add up To?

##### Age 11 to 18Challenge Level
If you take four consecutive numbers and add them together, the answer will always be even. What else do you notice?

### Impossible Sums

##### Age 14 to 18Challenge Level
Which numbers cannot be written as the sum of two or more consecutive numbers?

### Difference of Odd Squares

##### Age 14 to 18Challenge Level
$40$ can be written as $7^2 - 3^2.$ Which other numbers can be written as the difference of squares of odd numbers?

### Direct Logic

##### Age 16 to 18Challenge Level
Can you work through these direct proofs, using our interactive proof sorters?

### Adding Odd Numbers (part 2)

##### Age 16 to 18Challenge Level
Can you use Proof by Induction to establish what will happen when you add more and more odd numbers?

### KS5 Proof Shorts

##### Age 16 to 18Challenge Level
Here are a few questions taken from the Test of Mathematics for University Admission (or TMUA).

### Dodgy Proofs

##### Age 16 to 18Challenge Level
These proofs are wrong. Can you see why?

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