The key to solving these problems is to notice patterns or properties. Encouraging students to organise their work systematically allows them to notice what might not otherwise be obvious.

These problems challenge students to find all possible solutions. One of the best answers to "How do you know you have found them all" is to be able to say "I worked systematically!"

How many solutions can you find to this sum? Each of the different letters stands for a different number.

This problem offers a simple context for students to explore, make generalisations and prove conjectures, working numerically and algebraically.

Sometimes area and perimeter of rectangles are taught separately, and are often confused. In this problem students consider the relationship between them.

This problem is a good activity for the visualisation of symmetry, and for encouraging students to work systematically.

Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?

This problem encourages students to think about the properties of numbers. It could be used as an introduction to work on linear sequences and straight line graphs.

This problem encourages students to use coordinates, area and isosceles triangles to solve a non-standard problem. To find all possible solutions they will need to work systematically.

The engaging nature of this trick means that students are often prepared to persevere on this task. It may offer a chance to nurture a sense of resilience amongst your students.

This is an engaging context in which to reinforce rules of divisibility and challenge students to reason mathematically and work systematically.

This problem is inaccessible without looking at simpler cases, and thus helps students to see the value of specialising in order to generalise.

This problem offers the students an opportunity to consolidate what they are expected to know about mean, mode and median whilst also challenging them to work systematically, and justify their reasoning

This problem allows students to consolidate their understanding of how to calculate the area of irregular shapes, while offering an opportunity to explore and discover an interesting result.

You have 27 small cubes, 3 each of nine colours. Use the small cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of every colour.

This problem could be used to reinforce work on recording and describing linear sequences.

This problem could replace repetitive textbook work on calculating fractions of integers. It offers plenty of practice of these calculations while requiring students to come up with problem-solving strategies.

As well as introducing the difference of two squares, this problem allows students to explore, conjecture and use algebra to justify their results.

This problem provides an introduction to advanced mathematical behaviour which might not typically be encountered until university. The content level is secondary, but the thinking is sophisticated and will benefit the mathematical development of school-aged mathematicians.