List

Can you Prove it?

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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.

The last day for sending in your solutions to the live problems is Monday 31 January.

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

problem

Favourite

Three neighbours

Age

7 to 14

Challenge level

Take three consecutive numbers and add them together. What do you notice?

problem

Three consecutive odd numbers

Age

11 to 16

Challenge level

How many sets of three consecutive odd numbers can you find, in which all three numbers are prime?

problem

Adding odd numbers

Age

11 to 16

Challenge level

Is there a quick and easy way to calculate the sum of the first 100 odd numbers?

problem

Where are the primes?

Age

11 to 16

Challenge level

What can we say about all the primes which are greater than 3?

problem

What does it all add up to?

Age

11 to 18

Challenge level

If you take four consecutive numbers and add them together, the answer will always be even. What else do you notice?

problem

Different products

Age

14 to 16

Challenge level

Take four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

problem

Impossible sums

Age

14 to 18

Challenge level

Which numbers cannot be written as the sum of two or more consecutive numbers?

problem

Difference of odd squares

Age

14 to 18

Challenge level

$40$ can be written as $7^2 - 3^2.$ Which other numbers can be written as the difference of squares of odd numbers?

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