The key to solving these problems is to notice patterns or properties. Organising your work systematically allows you to notice what might not otherwise be obvious.

These problems challenge you 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.

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

Can you picture how to order the cards to reproduce Charlie's card trick for yourself?

Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...

If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?

Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?

Can you find rectangles where the value of the area is the same as the value of the perimeter?

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

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

How many different symmetrical shapes can you make by shading triangles or squares?

Can you find a way to identify times tables after they have been shifted up?

Whenever a monkey has peaches, he always keeps a fraction of them each day, gives the rest away, and then eats one. How long could he make his peaches last for?

Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?

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.

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

Try to solve this very difficult problem and then study our two suggested solutions. How would you use your knowledge to try to solve variants on the original problem?