Explaining, convincing and proving

problem

Euler meets Schlegel

Age

16 to 18

Challenge level

Discover how networks can be used to prove Euler's Polyhedron formula.
problem

Exponential intersection

Age

16 to 18

Challenge level

Can the pdfs and cdfs of an exponential distribution intersect?
problem

Rational roots

Age

16 to 18

Challenge level

Given that a, b and c are natural numbers show that if sqrt a+sqrt b is rational then it is a natural number. Extend this to 3 variables.
problem
Favourite

Square mean

Age

14 to 16

Challenge level

Is the mean of the squares of two numbers greater than, or less than, the square of their means?
problem

Diophantine n-tuples

Age

14 to 16

Challenge level

Can you explain why a sequence of operations always gives you perfect squares?
problem

DOTS division

Age

14 to 16

Challenge level

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.
problem

Target six

Age

16 to 18

Challenge level

Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.
problem

No right angle here

Age

14 to 16

Challenge level

Prove that the internal angle bisectors of a triangle will never be perpendicular to each other.
problem

Always the same

Age

11 to 14

Challenge level

Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number. Cross out the numbers on the same row and column. Repeat this process. Add up you four numbers. Why do they always add up to 34?
problem

More marbles

Age

11 to 14

Challenge level

I start with a red, a blue, a green and a yellow marble. I can trade any of my marbles for three others, one of each colour. Can I end up with exactly two marbles of each colour?