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?
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?
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. . . .
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?
How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?
A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work?
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?
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?
What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
What is the best way to shunt these carriages so that each train can continue its journey?
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?
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?
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 will you go about finding all the jigsaw pieces that have one peg and one hole?
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?
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?
How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?
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.
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?
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?
This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?
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?
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,. . . .
How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?
Imagine a wheel with different markings painted on it at regular intervals. Can you predict the colour of the 18th mark? The 100th mark?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?
Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?
How can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?
Is it possible to rearrange the numbers 1,2......12 around a clock face in such a way that every two numbers in adjacent positions differ by any of 3, 4 or 5 hours?
Can you predict when you'll be clapping and when you'll be clicking if you start this rhythm? How about when a friend begins a new rhythm at the same time?
Can you fit the tangram pieces into the outline of this goat and giraffe?
Can you work out what shape is made by folding in this way? Why not create some patterns using this shape but in different sizes?
Looking at the picture of this Jomista Mat, can you decribe what you see? Why not try and make one yourself?
Can you fit the tangram pieces into the outline of this sports car?
Can you fit the tangram pieces into the outline of this shape. How would you describe it?
Can you fit the tangram pieces into the outlines of the chairs?
Imagine you have six different colours of paint. You paint a cube using a different colour for each of the six faces. How many different cubes can be painted using the same set of six colours?
Here's a simple way to make a Tangram without any measuring or ruling lines.
Can you fit the tangram pieces into the outline of Little Ming?
On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight hidden squares?
Can you cut up a square in the way shown and make the pieces into a triangle?
One face of a regular tetrahedron is painted blue and each of the remaining faces are painted using one of the colours red, green or yellow. How many different possibilities are there?
Blue Flibbins are so jealous of their red partners that they will not leave them on their own with any other bue Flibbin. What is the quickest way of getting the five pairs of Flibbins safely to. . . .
Can you make a 3x3 cube with these shapes made from small cubes?
Can you fit the tangram pieces into the outline of Granma T?
Each of the nets of nine solid shapes has been cut into two pieces. Can you see which pieces go together?