Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?
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. . . .
Imagine a pyramid which is built in square layers of small cubes. If we number the cubes from the top, starting with 1, can you picture which cubes are directly below this first cube?
Make one big triangle so the numbers that touch on the small triangles add to 10. You could use the interactivity to help you.
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?
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?
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?
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?
Find your way through the grid starting at 2 and following these operations. What number do you end on?
Use the three triangles to fill these outline shapes. Perhaps you can create some of your own shapes for a friend to fill?
Move just three of the circles so that the triangle faces in the opposite direction.
Can you cover the camel with these pieces?
A variant on the game Alquerque
What happens when you try and fit the triomino pieces into these two grids?
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?
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?
Exchange the positions of the two sets of counters in the least possible number of moves
Can you make a 3x3 cube with these shapes made from small cubes?
A game for two players. You'll need some counters.
How many different triangles can you make on a circular pegboard that has nine pegs?
Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?
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?
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?
If you can post the triangle with either the blue or yellow colour face up, how many ways can it be posted altogether?
How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?
Imagine a wheel with different markings painted on it at regular intervals. Can you predict the colour of the 18th mark? The 100th mark?
We start with one yellow cube and build around it to make a 3x3x3 cube with red cubes. Then we build around that red cube with blue cubes and so on. How many cubes of each colour have we used?
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?
Take it in turns to place a domino on the grid. One to be placed horizontally and the other vertically. Can you make it impossible for your opponent to play?
This article for teachers describes how modelling number properties involving multiplication using an array of objects not only allows children to represent their thinking with concrete materials,. . . .
Investigate how the four L-shapes fit together to make an enlarged L-shape. You could explore this idea with other shapes too.
Design an arrangement of display boards in the school hall which fits the requirements of different people.
Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?
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?
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?
How can the same pieces of the tangram make this bowl before and after it was chipped? Use the interactivity to try and work out what is going on!
How many different ways can you find of fitting five hexagons together? How will you know you have found all the 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?
In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square?
Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper.
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 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?
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?
Which of these dice are right-handed and which are left-handed?
Can you fit the tangram pieces into the outlines of the watering can and man in a boat?
Here are shadows of some 3D shapes. What shapes could have made them?
A game for 2 players. Can be played online. One player has 1 red counter, the other has 4 blue. The red counter needs to reach the other side, and the blue needs to trap the red.
Can you fit the tangram pieces into the outline of this plaque design?
A shape and space game for 2,3 or 4 players. Be the last person to be able to place a pentomino piece on the playing board. Play with card, or on the computer.