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?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?
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?
There are lots of different methods to find out what the shapes are worth - how many can you find?
Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
Imagine a large cube made from small red cubes being dropped into a
pot of yellow paint. How many of the small cubes will have yellow
paint on their faces?
Can you find the values at the vertices when you know the values on
Is this a fair game? How many ways are there of creating a fair
game by adding odd and even numbers?
Can you describe this route to infinity? Where will the arrows take you next?
In this problem, we have created a pattern from smaller and smaller
squares. If we carried on the pattern forever, what proportion of
the image would be coloured blue?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
A collection of short Stage 3 and 4 problems on Reasoning, Justifying, Convincing and Proof.