Reasoning, convincing and proving

problem
Pythagorean fibs

Age

16 to 18

Challenge level

What have Fibonacci numbers got to do with Pythagorean triples?
problem
Fibonacci fashion

Age

16 to 18

Challenge level

What have Fibonacci numbers to do with solutions of the quadratic equation x^2 - x - 1 = 0 ?
problem
Diagonal sums

Age

7 to 14

Challenge level

In this 100 square, look at the green square which contains the numbers 2, 3, 12 and 13. What is the sum of the numbers that are diagonally opposite each other? What do you notice?
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
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.