How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?
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?
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.
Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper.
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?
How many different symmetrical shapes can you make by shading triangles or squares?
How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?
Use the clues about the symmetrical properties of these letters to place them on the grid.
An activity making various patterns with 2 x 1 rectangular tiles.
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?
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?
Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?
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?
What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
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.
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. . . .
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?
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?
Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?
These practical challenges are all about making a 'tray' and covering it with paper.
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?
What is the best way to shunt these carriages so that each train can continue its journey?
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?
A challenging activity focusing on finding all possible ways of stacking rods.
In how many ways can you stack these rods, following the rules?
This magic square has operations written in it, to make it into a maze. Start wherever you like, go through every cell and go out a total of 15!
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?
These are the faces of Will, Lil, Bill, Phil and Jill. Use the clues to work out which name goes with each face.
If we had 16 light bars which digital numbers could we make? How will you know you've found them all?
Using the statements, can you work out how many of each type of rabbit there are in these pens?
Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.
This activity investigates how you might make squares and pentominoes from Polydron.
Can you fill in this table square? The numbers 2 -12 were used to generate it with just one number used twice.
Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others.
Find the product of the numbers on the routes from A to B. Which route has the smallest product? Which the largest?
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?
Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?
Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?
The planet of Vuvv has seven moons. Can you work out how long it is between each super-eclipse?
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.
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?
This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!
In the planet system of Octa the planets are arranged in the shape of an octahedron. How many different routes could be taken to get from Planet A to Planet Zargon?
You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.
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?
Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?
Investigate the different ways you could split up these rooms so that you have double the number.
There are 4 jugs which hold 9 litres, 7 litres, 4 litres and 2 litres. Find a way to pour 9 litres of drink from one jug to another until you are left with exactly 3 litres in three of the jugs.
Place the numbers 1 to 8 in the circles so that no consecutive numbers are joined by a line.