Problems
Spiders and Flies
Stage: 1 Challenge Level: 
There were 22 legs creeping across the web. How many flies? How many spiders?
A City of Towers
Stage: 1 Challenge Level:
In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?
Tricky Triangles
Stage: 2 Challenge Level:
How many triangles are in this drawing? Name the different types of triangles you find. How many of each type of triangle can you find?
Pumpkin Pie Problem
Stage: 2 Challenge Level:
Peter wanted to make two pies for a party. His mother had a recipe for him to use. However, she always made 80 pies at a time. Did Peter have enough ingredients to make two pumpkin pies?
Creating Cubes
Stage: 2 Challenge Level:
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.
World of Tan - A Storm in a Tea Cup
Stage: 2 Challenge Level:
Can you fit the tangram pieces into the outline of these convex shapes?
Counting on Letters
Stage: 3 Challenge Level:
The letters of the word ABACUS have been arranged in the shape of a triangle. How many different ways can you find to read the word ABACUS from this triangular pattern?
Euromaths
Stage: 3 Challenge Level:
How many ways can you write the word EUROMATHS by starting at the top left hand corner and taking the next letter by stepping one step down or one step to the right in a 5x5 array?
Cross-country Race
Stage: 3 Challenge Level:
Eight children enter the autumn cross-country race at school. How many possible ways could they come in at first, second and third places?
Pairs
Stage: 3 Challenge Level:
Ann thought of 5 numbers and told Bob all the sums that could be made by adding the numbers in pairs. The list of sums is 6, 7, 8, 8, 9, 9, 10,10, 11, 12. Help Bob to find out which numbers Ann was. . . .
Funny Factorisation
Stage: 3 Challenge Level:
Some 4 digit numbers can be written as the product of a 3 digit number and a 2 digit number using the digits 1 to 9 each once and only once. The number 4396 can be written as just such a product. Can. . . .
LOGO Challenge - Tilings
Stage: 3 and 4 Challenge Level:
Three examples of particular tilings of the plane, namely those where - NOT all corners of the tile are vertices of the tiling. You might like to produce an elegant program to replicate one or all. . . .
Take a Square
Stage: 4 Challenge Level:
The diagonal of a square intersects the line joining one of the unused corners to the midpoint of the opposite side. What do you notice about the line segments produced?
What's Possible?
Stage: 4 Challenge Level:
Many numbers can be expressed as the difference of two perfect squares. For example 20 = 36 - 16 and 21 = 25 - 4. If the difference of the squares of two integers a and b is 924, what are the. . . .
Gold Again
Stage: 5 Challenge Level:
Without using a calculator, computer or tables find the exact values of cos36cos72 and also cos36 - cos72.
More Polynomial Equations
Stage: 5 Challenge Level:
Find relationships between the polynomials a, b and c which are polynomials in n giving the sums of the first n natural numbers, squares and cubes respectively.
Rational Roots
Stage: 5 Challenge Level:
Given that a, b and c are natural numbers show that if sqrt a+sqrt b is rational then it is a natural number. Extend this to 3 variables.
