Problems
What Shape and Colour?
Stage: 1 Challenge Level: 
Can you fill in the empty boxes in the grid with the right shape and colour?
2,4,6,8
Stage: 1 Challenge Level:
Using the cards 2, 4, 6, 8, +, - and =, what number statements can you make?
Bracelets
Stage: 2 Challenge Level:
Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?
Zios and Zepts
Stage: 2 Challenge Level:
On the planet Vuv there are two sorts of creatures. The Zios have 3 legs and the Zepts have 7 legs. The great planetary explorer Nico counted 52 legs. How many Zios and how many Zepts were there?
Neighbours
Stage: 2 Challenge Level:
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?
World of Tan - Animals
Stage: 2 Challenge Level:
Can you fit the tangram pieces into the outline of this goat and giraffe?
Waiting for Blast Off
Stage: 2 Challenge Level:
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?
Wag Worms
Stage: 2 Challenge Level:
When intergalactic Wag Worms are born they look just like a cube. Each year they grow another cube in any direction. Find all the shapes that five-year-old Wag Worms can be.
LOGO Challenge - Tiling with Equilateral Triangles
Stage: 3 Challenge Level:
At a maths conference someone said that - "not much can be done when tiling the plane with equilateral triangles." Below are just four tilings that grabbed my attention. Try replicating them in LOGO. . . .
Ones Only
Stage: 3 Challenge Level:
Find the smallest whole number which, when mutiplied by 7, gives a product consisting entirely of ones.
Please Explain
Stage: 3 Challenge Level:
Visitors to Earth from the distant planet of Zub-Zorna were amazed when they found out that when the digits in this multiplication were reversed, the answer was the same! Find a way to explain. . . .
Do Unto Caesar
Stage: 3 Challenge Level:
At the beginning of the night three poker players; Alan, Bernie and Craig had money in the ratios 7 : 6 : 5. At the end of the night the ratio was 6 : 5 : 4. One of them won $1 200. What were the. . . .
One Basket or Group Photo
Stage: 2, 3, 4 and 5 Challenge Level:
Libby Jared helped to set up NRICH and this is one of her favourite problems. It's a problem suitable for a wide age range and best tackled practically.
Zig Zag
Stage: 4 Challenge Level:
Four identical right angled triangles are drawn on the sides of a square. Two face out, two face in. Why do the four vertices marked with dots lie on one line?
Cyclic Quad Jigsaw
Stage: 4 Challenge Level:
A picture is made by joining five small quadrilaterals together to make a large quadrilateral. Is it possible to draw a similar picture if all the small quadrilaterals are cyclic?
The Root Cause
Stage: 5 Challenge Level:
Prove that if a is a natural number and the square root of a is rational, then it is a square number (an integer n^2 for some integer n.)
Good Approximations
Stage: 5 Challenge Level:
Solve quadratic equations and use continued fractions to find rational approximations to irrational numbers.
Biggest Bendy
Stage: 5 Challenge Level:
Four rods are hinged at their ends to form a quadrilateral with fixed side lengths. Show that the quadrilateral has a maximum area when it is cyclic.
