Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

This task encourages you to investigate the number of edging pieces and panes in different sized windows.

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?

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

Six friends sat around a circular table. Can you work out from the information who sat where and what their profession were?

There are seven pots of plants in a greenhouse. They have lost their labels. Perhaps you can help re-label them.

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'.

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

Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.

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

Use the clues to work out which cities Mohamed, Sheng, Tanya and Bharat live in.

The Vikings communicated in writing by making simple scratches on wood or stones called runes. Can you work out how their code works using the table of the alphabet?

You have been given nine weights, one of which is slightly heavier than the rest. Can you work out which weight is heavier in just two weighings of the balance?

This task, written for the National Young Mathematicians' Award 2016, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . .

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?

A tetromino is made up of four squares joined edge to edge. Can this tetromino, together with 15 copies of itself, be used to cover an eight by eight chessboard?

These activities focus on finding all possible solutions so if you work in a systematic way, you won't leave any out.

Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?

The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

Can you find six numbers to go in the Daisy from which you can make all the numbers from 1 to a number bigger than 25?

10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways?

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.

Sitting around a table are three girls and three boys. Use the clues to work out were each person is sitting.

What is the best way to shunt these carriages so that each train can continue its journey?

Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?

The Zargoes use almost the same alphabet as English. What does this birthday message say?

What is the smallest number of jumps needed before the white rabbits and the grey rabbits can continue along their path?

A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work?

Use the clues to find out who's who in the family, to fill in the family tree and to find out which of the family members are mathematicians and which are not.

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.

Can you create jigsaw pieces which are based on a square shape, with at least one peg and one hole?

Replace the letters with numbers to make the addition work out correctly. R E A D + T H I S = P A G E

Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?

You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?

Hover your mouse over the counters to see which ones will be removed. Click to remove them. The winner is the last one to remove a counter. How you can make sure you win?

A dog is looking for a good place to bury his bone. Can you work out where he started and ended in each case? What possible routes could he have taken?

Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?

Make a pair of cubes that can be moved to show all the days of the month from the 1st to the 31st.

Make your own double-sided magic square. But can you complete both sides once you've made the pieces?

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?

These activities lend themselves to systematic working in the sense that it helps if you have an ordered approach.

Can you work out how many cubes were used to make this open box? What size of open box could you make if you had 112 cubes?

Design an arrangement of display boards in the school hall which fits the requirements of different people.

What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?

These practical challenges are all about making a 'tray' and covering it with paper.

Nina must cook some pasta for 15 minutes but she only has a 7-minute sand-timer and an 11-minute sand-timer. How can she use these timers to measure exactly 15 minutes?

Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

A merchant brings four bars of gold to a jeweller. How can the jeweller use the scales just twice to identify the lighter, fake bar?

These activities lend themselves to systematic working in the sense that it helps to have an ordered approach.