Reasoning, convincing and proving

problem
Break it up!

Age

5 to 11

Challenge level

In how many different ways can you break up a stick of seven interlocking cubes? Now try with a stick of eight cubes and a stick of six cubes. What do you notice?
problem
More number pyramids

Age

11 to 14

Challenge level

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
problem
Odd squares

Age

7 to 11

Challenge level

Think of a number, square it and subtract your starting number. Is the number you're left with odd or even? How do the images help to explain this?
problem
Generally geometric

Age

16 to 18

Challenge level

Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.
problem
Circle box

Age

14 to 16

Challenge level

It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?
problem
An unusual shape

Age

11 to 14

Challenge level

Can you maximise the area available to a grazing goat?
problem
Special numbers

Age

11 to 14

Challenge level

My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?
problem
Triangles and petals

Age

14 to 16

Challenge level

An equilateral triangle rotates around regular polygons and produces an outline like a flower. What are the perimeters of the different flowers?
problem
Flexi quad tan

Age

16 to 18

Challenge level

As a quadrilateral Q is deformed (keeping the edge lengths constnt) the diagonals and the angle X between them change. Prove that the area of Q is proportional to tanX.
problem
Polite numbers

Age

16 to 18

Challenge level

A polite number can be written as the sum of two or more consecutive positive integers, for example 8+9+10=27 is a polite number. Can you find some more polite, and impolite, numbers?