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?
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?
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?
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?
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?
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?
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?
Design an arrangement of display boards in the school hall which fits the requirements of different people.
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?
My cousin was 24 years old on Friday April 5th in 1974. On what day of the week was she born?
How many triangles can you make using sticks that are 3cm, 4cm and 5cm long?
How many different triangles can you draw on the dotty grid which each have one dot in the middle?
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?
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?
Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper.
How many different rhythms can you make by putting two drums on the wheel?
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 work out how to balance this equaliser? You can put more than one weight on a hook.
Use the clues to colour each square.
How many different triangles can you make on a circular pegboard that has nine pegs?
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 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 triangles can you make on the 3 by 3 pegboard?
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?
Find your way through the grid starting at 2 and following these operations. What number do you end on?
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.?
An activity making various patterns with 2 x 1 rectangular tiles.
The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?
These practical challenges are all about making a 'tray' and covering it with paper.
How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?
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?
Can you cover the camel with these pieces?
What happens when you try and fit the triomino pieces into these two grids?
Place eight dots on this diagram, so that there are only two dots on each straight line and only two dots on each circle.
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?
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?
Place the numbers 1 to 6 in the circles so that each number is the difference between the two numbers just below it.
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.
Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?
Imagine that the puzzle pieces of a jigsaw are roughly a rectangular shape and all the same size. How many different puzzle pieces could there be?
Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.
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?
Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?
Can you find all the different triangles on these peg boards, and find their angles?
Alice's mum needs to go to each child's house just once and then back home again. How many different routes are there? Use the information to find out how long each road is on the route she took.
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?