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.
A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?
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 can you put five cereal packets together to make different shapes if you must put them face-to-face?
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?
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 smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
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?
Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!
These activities focus on finding all possible solutions so if you work in a systematic way, you won't leave any out.
Nina must cook some pasta for 15 minutes but she only has a 7-minute sand-timer and an 11-minute sand-timer. How can she use these timers to measure exactly 15 minutes?
Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?
These activities lend themselves to systematic working in the sense that it helps to have an ordered approach.
Make a pair of cubes that can be moved to show all the days of the month from the 1st to the 31st.
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.
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?
Use the clues to work out which cities Mohamed, Sheng, Tanya and Bharat live in.
These activities focus on finding all possible solutions so working in a systematic way will ensure none are left out.
Alice and Brian are snails who live on a wall and can only travel along the cracks. Alice wants to go to see Brian. How far is the shortest route along the cracks? Is there more than one way to go?
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?
How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?
Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?
These are the faces of Will, Lil, Bill, Phil and Jill. Use the clues to work out which name goes with each face.
These activities lend themselves to systematic working in the sense that it helps if you have an ordered approach.
Can you create jigsaw pieces which are based on a square shape, with at least one peg and one hole?
Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?
The Zargoes use almost the same alphabet as English. What does this birthday message say?
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?
When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?
My dice has inky marks on each face. Can you find the route it has taken? What does each face look like?
Sitting around a table are three girls and three boys. Use the clues to work out were each person is sitting.
What is the best way to shunt these carriages so that each train can continue its journey?
If we had 16 light bars which digital numbers could we make? How will you know you've found them all?
A merchant brings four bars of gold to a jeweller. How can the jeweller use the scales just twice to identify the lighter, fake bar?
Six friends sat around a circular table. Can you work out from the information who sat where and what their profession were?
How many models can you find which obey these rules?
Seven friends went to a fun fair with lots of scary rides. They decided to pair up for rides until each friend had ridden once with each of the others. What was the total number rides?
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'.
You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.
What is the smallest number of jumps needed before the white rabbits and the grey rabbits can continue along their path?
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 find all the ways to get 15 at the top of this triangle of numbers? Many opportunities to work in different ways.
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 many cubes were used to make this open box? What size of open box could you make if you had 112 cubes?
If you have three circular objects, you could arrange them so that they are separate, touching, overlapping or inside each other. Can you investigate all the different possibilities?
What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?
In the planet system of Octa the planets are arranged in the shape of an octahedron. How many different routes could be taken to get from Planet A to Planet Zargon?
How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?