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?
Make your own double-sided magic square. But can you complete both sides once you've made the pieces?
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?
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?
Whenever a monkey has peaches, he always keeps a fraction of them each day, gives the rest away, and then eats one. How long could he make his peaches last for?
Hover your mouse over the counters to see which ones will be removed. Click to remover them. The winner is the last one to remove a counter. How you can make sure you win?
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?
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?
Design an arrangement of display boards in the school hall which fits the requirements of different people.
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?
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?
What is the best way to shunt these carriages so that each train can continue its journey?
Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.
Find the product of the numbers on the routes from A to B. Which route has the smallest product? Which the largest?
Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?
Can you find all the different ways of lining up these Cuisenaire rods?
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?
This task, written for the National Young Mathematicians' Award 2016, invites you to explore the different combinations of scores that you might get on these dart boards.
A mathematician goes into a supermarket and buys four items. Using a calculator she multiplies the cost instead of adding them. How can her answer be the same as the total at the till?
How many different triangles can you make on a circular pegboard that has nine pegs?
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....?
What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?
The planet of Vuvv has seven moons. Can you work out how long it is between each super-eclipse?
How many trapeziums, of various sizes, are hidden in this picture?
There is a long tradition of creating mazes throughout history and across the world. This article gives details of mazes you can visit and those that you can tackle on paper.
Can you rearrange the biscuits on the plates so that the three biscuits on each plate are all different and there is no plate with two biscuits the same as two biscuits on another plate?
You have two egg timers. One takes 4 minutes exactly to empty and the other takes 7 minutes. What times in whole minutes can you measure and how?
Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?
On a digital 24 hour clock, at certain times, all the digits are consecutive. How many times like this are there between midnight and 7 a.m.?
Sitting around a table are three girls and three boys. Use the clues to work out were each person is sitting.
Seven friends went to a fun fair with lots of scary rides. They decided to pair up for rides until each friend had ridden once with each of the others. What was the total number rides?
Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles?
The Zargoes use almost the same alphabet as English. What does this birthday message say?
When intergalactic Wag Worms are born they look just like a cube. Each year they grow another cube in any direction. Find all the shapes that five-year-old Wag Worms can be.
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.
Put 10 counters in a row. Find a way to arrange the counters into five pairs, evenly spaced in a row, in just 5 moves, using the rules.
How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?
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?
Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.
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. . . .
A few extra challenges set by some young NRICH members.
How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.
Can you fill in this table square? The numbers 2 -12 were used to generate it with just one number used twice.
Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.
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?
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.
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?
The letters of the word ABACUS have been arranged in the shape of a triangle. How many different ways can you find to read the word ABACUS from this triangular pattern?
These are the faces of Will, Lil, Bill, Phil and Jill. Use the clues to work out which name goes with each face.
Using the statements, can you work out how many of each type of rabbit there are in these pens?