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.

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

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?

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?

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

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.

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?

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?

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

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?

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?

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?

How will you go about finding all the jigsaw pieces that have one peg and one hole?

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?

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

Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

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

Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

How many different triangles can you make on a circular pegboard that has nine pegs?

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

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?

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?

How many solutions can you find to this sum? Each of the different letters stands for a different number.

The letters in the following addition sum represent the digits 1 ... 9. If A=3 and D=2, what number is represented by "CAYLEY"?

Two sudokus in one. Challenge yourself to make the necessary connections.

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'.

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

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

Two sudokus in one. Challenge yourself to make the necessary connections.

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?

You need to find the values of the stars before you can apply normal Sudoku rules.

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?

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

A game for 2 people. Take turns placing a counter on the star. You win when you have completed a line of 3 in your colour.

Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

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

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. . . .

This second Sudoku article discusses "Corresponding Sudokus" which are pairs of Sudokus with terms that can be matched using a substitution rule.

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

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

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?

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.

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.

This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.

Bellringers have a special way to write down the patterns they ring. Learn about these patterns and draw some of your own.

Mr McGregor has a magic potting shed. Overnight, the number of plants in it doubles. He'd like to put the same number of plants in each of three gardens, planting one garden each day. Can he do it?

A few extra challenges set by some young NRICH members.