This article introduces the idea of generic proof for younger children and illustrates how one example can offer a proof of a general result through unpacking its underlying structure.
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?
Imagine a wheel with different markings painted on it at regular intervals. Can you predict the colour of the 18th mark? The 100th mark?
Think of a number, square it and subtract your starting number. Is the number you’re left with odd or even? How do the images help to explain this?
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,. . . .
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 remove them. The winner is the last one to remove a counter. How you can make sure you win?
A game for 1 or 2 people. Use the interactive version, or play with friends. Try to round up as many counters as possible.
Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.
Can you find a cuboid that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?
Design an arrangement of display boards in the school hall which fits the requirements of different people.
Seeing Squares game for an adult and child. Can you come up with a way of always winning this game?
What is the best way to shunt these carriages so that each train can continue its journey?
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. . . .
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 game for 1 person. Can you work out how the dice must be rolled from the start position to the finish? Play on line.
A and B are two interlocking cogwheels having p teeth and q teeth respectively. One tooth on B is painted red. Find the values of p and q for which the red tooth on B contacts every gap on the. . . .
How will you go about finding all the jigsaw pieces that have one peg and one hole?
This article for teachers discusses examples of problems in which there is no obvious method but in which children can be encouraged to think deeply about the context and extend their ability to. . . .
When I fold a 0-20 number line, I end up with 'stacks' of numbers on top of each other. These challenges involve varying the length of the number line and investigating the 'stack totals'.
This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?
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?
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
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?
Can you discover whether this is a fair game?
How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?
Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?
How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?
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?
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?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?
How many different triangles can you make on a circular pegboard that has nine pegs?
For this task, you'll need an A4 sheet and two A5 transparent sheets. Decide on a way of arranging the A5 sheets on top of the A4 sheet and explore ...
When dice land edge-up, we usually roll again. But what if we didn't...?
How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?
Have a go at this 3D extension to the Pebbles problem.
Can you make a 3x3 cube with these shapes made from small cubes?
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?
This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.
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 find a way of counting the spheres in these arrangements?
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
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 standard die has the numbers 1, 2 and 3 are opposite 6, 5 and 4 respectively so that opposite faces add to 7? If you make standard dice by writing 1, 2, 3, 4, 5, 6 on blank cubes you will find. . . .
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?
Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
A game for two players on a large squared space.