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

This dice train has been made using specific rules. How many different trains can you make?

This task depends on groups working collaboratively, discussing and reasoning to agree a final product.

How could you arrange at least two dice in a stack so that the total of the visible spots is 18?

Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.

Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?

Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?

Can you draw a square in which the perimeter is numerically equal to the area?

When I fold a 0-20 number line, I end up with 'stacks' of numbers on top of each other. These challenges involve varying the length of the number line and investigating the 'stack totals'.

These rectangles have been torn. How many squares did each one have inside it before it was ripped?

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?

Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

This activity investigates how you might make squares and pentominoes from Polydron.

In this challenge, buckets come in five different sizes. If you choose some buckets, can you investigate the different ways in which they can be filled?

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

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

Add or subtract the two numbers on the spinners and try to complete a row of three. Are there some numbers that are good to aim for?

How can you put five cereal packets together to make different shapes if you must put them face-to-face?

Use the clues about the symmetrical properties of these letters to place them on the grid.

How many models can you find which obey these rules?

How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?

This article for primary teachers suggests ways in which to help children become better at working systematically.

When you throw two regular, six-faced dice you have more chance of getting one particular result than any other. What result would that be? Why is this?

Here are four cubes joined together. How many other arrangements of four cubes can you find? Can you draw them on dotty paper?

If you had any number of ordinary dice, what are the possible ways of making their totals 6? What would the product of the dice be each time?

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

Systematically explore the range of symmetric designs that can be created by shading parts of the motif below. Use normal square lattice paper to record your results.

The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.

Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?

How many rectangles can you find in this shape? Which ones are differently sized and which are 'similar'?

What happens when you round these numbers to the nearest whole number?

An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?

Roll two red dice and a green dice. Add the two numbers on the red dice and take away the number on the green. What are all the different possible answers?

If you had 36 cubes, what different cuboids could you make?

Four numbers on an intersection that need to be placed in the surrounding cells. That is all you need to know to solve this sudoku.

Here you see the front and back views of a dodecahedron. Each vertex has been numbered so that the numbers around each pentagonal face add up to 65. Can you find all the missing numbers?

There were chews for 2p, mini eggs for 3p, Chocko bars for 5p and lollypops for 7p in the sweet shop. What could each of the children buy with their money?

Find out what a "fault-free" rectangle is and try to make some of your own.

Find out about Magic Squares in this article written for students. Why are they magic?!

Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?

Tim had nine cards each with a different number from 1 to 9 on it. How could he have put them into three piles so that the total in each pile was 15?

An investigation that gives you the opportunity to make and justify predictions.

This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.

The puzzle can be solved with the help of small clue-numbers which are either placed on the border lines between selected pairs of neighbouring squares of the grid or placed after slash marks on. . . .

Alice's mum needs to go to each child's house just once and then back home again. How many different routes are there? Use the information to find out how long each road is on the route she took.

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

Number problems at primary level that require careful consideration.

In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?