How many models can you find which obey these rules?
If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?
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?
What is the best way to shunt these carriages so that each train can continue its journey?
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?
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?
Design an arrangement of display boards in the school hall which fits the requirements of different people.
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?
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?
Here are some rods that are different colours. How could I make a dark green rod using yellow and white rods?
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. . . .
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?
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?
How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?
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 problem focuses on Dienes' Logiblocs. What is the same and what is different about these pairs of shapes? Can you describe the shapes in the picture?
Can you work out how to balance this equaliser? You can put more than one weight on a hook.
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?
Use the clues to colour each square.
Use the interactivity to help get a feel for this problem and to find out all the possible ways the balls could land.
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?
What happens when you try and fit the triomino pieces into these two grids?
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?
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?
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?
An activity making various patterns with 2 x 1 rectangular tiles.
Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper.
Can you cover the camel with these pieces?
Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?
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.
These practical challenges are all about making a 'tray' and covering it with paper.
Find your way through the grid starting at 2 and following these operations. What number do you end on?
How many different rhythms can you make by putting two drums on the wheel?
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.
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?
Here are four cubes joined together. How many other arrangements of four cubes can you find? Can you draw them on dotty paper?
Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.
Place the numbers 1 to 6 in the circles so that each number is the difference between the two numbers just below it.
How many different triangles can you draw on the dotty grid which each have one dot in the middle?
Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?
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.
There are to be 6 homes built on a new development site. They could be semi-detached, detached or terraced houses. How many different combinations of these can you find?
This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?
These two group activities use mathematical reasoning - one is numerical, one geometric.
Kate has eight multilink cubes. She has two red ones, two yellow, two green and two blue. She wants to fit them together to make a cube so that each colour shows on each face just once.
Place eight dots on this diagram, so that there are only two dots on each straight line and only two dots on each circle.
Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?
Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others.
Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.
In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?