Problems
Hand Span
Stage: 1 Challenge Level: 
Use your hand span to measure the distance around a tree trunk. If you ask a friend to try the same thing, how do the answers compare?
The Tomato and the Bean
Stage: 1 Challenge Level:
At the beginning of May Tom put his tomato plant outside. On the same day he sowed a bean in another pot. When will the two be the same height?
Lawn Border
Stage: 1 and 2 Challenge Level:
If I use 12 green tiles to represent my lawn, how many different ways could I arrange them? How many border tiles would I need each time?
Double Your Popcorn, Double Your Pleasure
Stage: 2 Challenge Level:
We went to the cinema and decided to buy some bags of popcorn so we asked about the prices. Investigate how much popcorn each bag holds so find out which we might have bought.
What's My Weight?
Stage: 2 Short Challenge Level:
There are four equal weights on one side of the scale and an apple on the other side. What can you say that is true about the apple and the weights from the picture?
Augustus' Age
Stage: 2 Challenge Level:
In 1871 a mathematician called Augustus De Morgan died. De Morgan made a puzzling statement about his age. Can you discover which year De Morgan was born in?
World of Tan - Clocks
Stage: 2 Challenge Level:
Can you fit the tangram pieces into the outlines of these clocks?
Four Goodness Sake
Stage: 2 Challenge Level:
Use 4 four times with simple operations so that you get the answer 12. Can you make 15, 16 and 17 too?
The Hare and the Tortoise
Stage: 2 Challenge Level:
In this version of the story of the hare and the tortoise, the race is 10 kilometres long. Can you work out how long the hare sleeps for using the information given?
Tis Unique
Stage: 3 Challenge Level:
This addition sum uses all ten digits 0, 1, 2...9 exactly once. Find the sum and show that the one you give is the only possibility.
Blue and White
Stage: 3 Challenge Level:
In the four examples below identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
Nonagon Tiling
Stage: 3 Challenge Level:
Several stimulating tilings of the plane can be obtained by considering extensions and modifications to the following procedure: TO TILE :N FD :N REPEAT 3 [RT 15 FD :N] LT 120 FD :N LT 75 FD :N. . . .
Even So
Stage: 3 Challenge Level:
Find some triples of whole numbers a, b and c such that a^2 + b^2 + c^2 is a multiple of 4. Is it necessarily the case that a, b and c must all be even? If so, can you explain why?
One O Five
Stage: 3 Challenge Level:
You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .
Novemberish
Stage: 4 Challenge Level:
a) A four digit number (in base 10) aabb is a perfect square. Discuss ways of systematically finding this number. (b) Prove that 11^{10}-1 is divisible by 100.
A Biggy
Stage: 4 Challenge Level:
Find the smallest positive integer N such that N/2 is a perfect cube, N/3 is a perfect fifth power and N/5 is a perfect seventh power.
Garfield's Proof
Stage: 4 Challenge Level:
Rotate a copy of the trapezium about the centre of the longest side of the blue triangle to make a square. Find the area of the square and then derive a formula for the area of the trapezium.
Factoring a Million
Stage: 4 Challenge Level:
In how many ways can the number 1 000 000 be expressed as the product of three positive integers?
Reach for Polydron
Stage: 5 Challenge Level:
In a tetrahedron two of the faces are identical equilateral triangles - side length 1 unit and two are right angled isosceles triangles. Find the exact volume of the tetrahedron.
Strange Rectangle
Stage: 5 Challenge Level:
ABCD is a rectangle and P, Q, R and S are moveable points on the edges dividing the edges in certain ratios. Strangely PQRS is always a cyclic quadrilateral and you can find the angles.
