Bang's Theorem

If all the faces of a tetrahedron have the same perimeter then show that they are all congruent.

Rudolff's Problem

A group of 20 people pay a total of £20 to see an exhibition. The admission price is £3 for men, £2 for women and 50p for children. How many men, women and children are there in the group?

Our Ages

I am exactly n times my daughter's age. In m years I shall be exactly (n-1) times her age. In m2 years I shall be exactly (n-2) times her age. After that I shall never again be an exact multiple of her age. Ages, n and m are all whole numbers. How old am I? Now suppose there is some wishful thinking in the above assertion and I have to admit to being older, and indeed that I will be an exact multiple of her age in m3 years. How old does this make me?

Root to Poly

Stage: 4 Challenge Level:

Why do this problem?
It gives learners experience of algebraic manipulation of polynomials and working with surds. It is based on the fact that if you know one root of a polynomial then you know one of its factors.

Possible approach
This can be used as a lesson starter.

Key questions
We are looking for a polynomial in $x$, do you know any values of $x$ that satisfy the polynomial?

If you have an expression involving surds what can you try in order to get rid of the square roots?