Reasoning, convincing and proving

problem
More numbers in the ring

Age

5 to 7

Challenge level

If there are 3 squares in the ring, can you place three different numbers in them so that their differences are odd? Try with different numbers of squares around the ring. What do you notice?
problem
Ring a ring of numbers

Age

5 to 7

Challenge level

Choose four of the numbers from 1 to 9 to put in the squares so that the differences between joined squares are odd.
problem
Triangles within pentagons

Age

14 to 16

Challenge level

Show that all pentagonal numbers are one third of a triangular number.
problem
Triangles within squares

Age

14 to 16

Challenge level

Can you find a rule which relates triangular numbers to square numbers?
problem
Triangles within triangles

Age

14 to 16

Challenge level

Can you find a rule which connects consecutive triangular numbers?
problem
Cuisenaire counting

Age

5 to 7

Challenge level

Here are some rods that are different colours. How could I make a yellow rod using white and red rods?
problem
Cubestick

Age

16 to 18

Challenge level

Stick some cubes together to make a cuboid. Find two of the angles by as many different methods as you can devise.
problem
Golden eggs

Age

16 to 18

Challenge level

Find a connection between the shape of a special ellipse and an infinite string of nested square roots.
problem
What's a group?

Age

16 to 18

Challenge level

Explore the properties of some groups such as: The set of all real numbers excluding -1 together with the operation x*y = xy + x + y. Find the identity and the inverse of the element x.
problem
Cyclic triangles

Age

16 to 18

Challenge level

Make and prove a conjecture about the cyclic quadrilateral inscribed in a circle of radius r that has the maximum perimeter and the maximum area.