Learning to Generalise

When working on maths problems, it's often a good idea to start by trying out some examples using small numbers, and looking for patterns, before looking for general rules. In this set of tasks, we invite you to move from particular cases to generalisations, and use the power of algebra to prove your findings.

Multiple Surprises and Square Number Surprises include solutions that have previously been submitted to NRICH, so you may wish to try these problems first and then compare your approach with the published ones. Then try Tilted Squares and Difference of Two Squares, and send us your solutions!

Multiple Surprises

Age 11 to 16 Challenge Level:

Sequences of multiples keep cropping up...

Square Number Surprises

Age 14 to 16 Challenge Level:

There are unexpected discoveries to be made about square numbers...

Tilted Squares

Age 11 to 14 Challenge Level:

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

Difference of Two Squares

Age 14 to 16 Challenge Level:

What is special about the difference between squares of numbers adjacent to multiples of three?