Can You Prove It?

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.
 

Three Neighbours 

Age 7 to 14
Challenge Level
Take three consecutive numbers and add them together. What do you notice?

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?

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?

Where Are the Primes? 

Age 11 to 16
Challenge Level
What can we say about all the primes which are greater than 3?

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?

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?

Impossible Sums 

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

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.