Creating and manipulating expressions and formulae

problem
How many miles to go?

Age

11 to 14

Challenge level

How many more miles must the car travel before the numbers on the milometer and the trip meter contain the same digits in the same order?
problem
Mind reading

Age

11 to 14

Challenge level

Think of a number, add one, double it, take away 3, add the number you first thought of, add 7, divide by 3 and take away the number you first thought of. You should now be left with 2. How do I know?
problem
Leonardo's problem

Age

14 to 18

Challenge level

A, B & C own a half, a third and a sixth of a coin collection. Each grab some coins, return some, then share equally what they had put back, finishing with their own share. How rich are they?
problem
Sums of squares

Age

16 to 18

Challenge level

Can you prove that twice the sum of two squares always gives the sum of two squares?
problem
Pick's theorem

Age

14 to 16

Challenge level

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.
problem
Top-heavy pyramids

Age

11 to 14

Challenge level

Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.
problem
Terminology

Age

14 to 16

Challenge level

Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
problem
Snookered

Age

14 to 18

Challenge level

In a snooker game the brown ball was on the lip of the pocket but it could not be hit directly as the black ball was in the way. How could it be potted by playing the white ball off a cushion?
problem
Algebra from geometry

Age

11 to 16

Challenge level

Account of an investigation which starts from the area of an annulus and leads to the formula for the difference of two squares.
problem
Think of two numbers

Age

11 to 14

Challenge level

Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?