Create a pattern on the left-hand grid. How could you extend your pattern on the right-hand grid?

There are three versions of this challenge. The idea is to change the colour of all the spots on the grid. Can you do it in fewer throws of the dice?

The computer has made a rectangle and will tell you the number of spots it uses in total. Can you find out where the rectangle is?

Three children are going to buy some plants for their birthdays. They will plant them within circular paths. How could they do this?

Choose 13 spots on the grid. Can you work out the scoring system? What is the maximum possible score?

Can you use the information to find out which cards I have used?

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...

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?

How can you change the area of a shape but keep its perimeter the same? How can you change the perimeter but keep the area the same?

How can you change the surface area of a cuboid but keep its volume the same? How can you change the volume but keep the surface area the same?

Alison and Charlie are playing a game. Charlie wants to go first so Alison lets him. Was that such a good idea?

Articles about mathematics which can help to invigorate your classroom

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?

The NRICH Stage 5 weekly challenges are shorter problems aimed at Post-16 students or enthusiastic younger students. There are 52 of them.

Can you find some Pythagorean Triples where the two smaller numbers differ by 1?

Invent shapes with different numbers of stable and unstable equilibrium points

Use graphs to gain insights into an area and perimeter problem, or use your knowledge of area and perimeter of rectangles to gain insight into the graphs.

We were overwhelmed with excellent solutions to this challenge. You gave very clear explanations to each different part of the problem.

Working systematically and introducing some algebra led to some excellent solutions to this problem.

In this article, the NRICH team describe the process of selecting solutions for publication on the site.

An article describing what LTHC tasks are, and why we think they're a good idea.

In this article Liz Woodham reflects on just how much we really listen to learners’ own questions to determine the mathematical path of lessons.

Maths is everywhere in the world! Take a look at these images. What mathematics can you see?