Explaining, convincing and proving

problem
Arrowhead

Age

14 to 16

Challenge level

The points P, Q, R and S are the midpoints of the edges of a non-convex quadrilateral.What do you notice about the quadrilateral PQRS and its area?
problem
Janine's conjecture

Age

14 to 16

Challenge level

Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. Does this always work? Can you prove or disprove this conjecture?
problem
Quad in quad

Age

14 to 18

Challenge level

Join the midpoints of a quadrilateral to get a new quadrilateral. What is special about it?
problem
Fixing the odds

Age

14 to 16

Challenge level

You have two bags, four red balls and four white balls. You must put all the balls in the bags although you are allowed to have one bag empty. How should you distribute the balls between the two bags so as to make the probability of choosing a red ball as small as possible and what will the probability be in that case?
problem
Tri-colour

Age

11 to 14

Challenge level

Six points are arranged in space so that no three are collinear. How many line segments can be formed by joining the points in pairs?
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?
problem
Find the fake

Age

14 to 16

Challenge level

There are 12 identical looking coins, one of which is a fake. The counterfeit coin is of a different weight to the rest. What is the minimum number of weighings needed to locate the fake coin?
problem
Legs eleven

Age

11 to 14

Challenge level

Take any four digit number. Move the first digit to the end and move the rest along. Now add your two numbers. Did you get a multiple of 11?
problem
Marbles

Age

11 to 14

Challenge level

I start with a red, a green and a blue marble. I can trade any of my marbles for two others, one of each colour. Can I end up with five more blue marbles than red after a number of such trades?
problem
Compare areas

Age

14 to 16

Challenge level

Which has the greatest area, a circle or a square, inscribed in an isosceles right angle triangle?