How many triangles can you make on the 3 by 3 pegboard?
Can you order pictures of the development of a frog from frogspawn and of a bean seed growing into a plant?
Design an arrangement of display boards in the school hall which fits the requirements of different people.
Pentagram Pylons - can you elegantly recreate them? Or, the European flag in LOGO - what poses the greater problem?
Can you recreate these designs? What are the basic units? What movement is required between each unit? Some elegant use of procedures will help - variables not essential.
Explore this how this program produces the sequences it does. What are you controlling when you change the values of the variables?
How will you go about finding all the jigsaw pieces that have one peg and one hole?
How can you put five cereal packets together to make different shapes if you must put them face-to-face?
Just four procedures were used to produce a design. How was it done? Can you be systematic and elegant so that someone can follow your logic?
Make your own double-sided magic square. But can you complete both sides once you've made the pieces?
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?
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?
What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
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?
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?
How many models can you find which obey these rules?
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. . . .
Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.
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?
Find all the different shapes that can be made by joining five equilateral triangles edge to edge.
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?
Time for a little mathemagic! Choose any five cards from a pack and show four of them to your partner. How can they work out the fifth?
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?
This activity investigates how you might make squares and pentominoes from Polydron.
Sitting around a table are three girls and three boys. Use the clues to work out were each person is sitting.
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 focus on finding all possible solutions so working in a systematic way will ensure none are left out.
Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?
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?
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?
The Zargoes use almost the same alphabet as English. What does this birthday message say?
What is the smallest number of jumps needed before the white rabbits and the grey rabbits can continue along their path?
Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?
Can you create jigsaw pieces which are based on a square shape, with at least one peg and one hole?
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?
Kate has eight multilink cubes. She has two red ones, two yellow, two green and two blue. She wants to fit them together to make a cube so that each colour shows on each face just once.
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?
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?
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?
Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.
Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!
If we had 16 light bars which digital numbers could we make? How will you know you've found them all?
Can you use your powers of logic and deduction to work out the missing information in these sporty situations?
These are the faces of Will, Lil, Bill, Phil and Jill. Use the clues to work out which name goes with each face.
If you had 36 cubes, what different cuboids could you make?
Make a pair of cubes that can be moved to show all the days of the month from the 1st to the 31st.
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?
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?
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.
These activities lend themselves to systematic working in the sense that it helps to have an ordered approach.