Reasoning, convincing and proving

problem
Subtended angles

Age

11 to 14

Challenge level

What is the relationship between the angle at the centre and the angles at the circumference, for angles which stand on the same arc? Can you prove it?
problem
Triangles in circles

Age

11 to 14

Challenge level

Can you find triangles on a 9-point circle? Can you work out their angles?
problem
More total totality

Age

11 to 14

Challenge level

Is it possible to arrange the numbers 1-6 on the nodes of this diagram, so that all the sums between numbers on adjacent nodes are different?
problem
Total totality

Age

11 to 14

Challenge level

Is it possible to arrange the numbers 1-6 on the nodes of this diagram, so that all the sums between numbers on adjacent nodes are different?
problem
Day of the Triffids

Age

14 to 16

Challenge level

Jasmine buys three different types of plant. How many triffids did she buy?
problem
Multiplication square

Age

14 to 16

Challenge level

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
problem
Plus or minus

Age

16 to 18

Challenge level

Make and prove a conjecture about the value of the product of the Fibonacci numbers $F_{n+1}F_{n-1}$.
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?