A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?

A and C are the opposite vertices of a square ABCD, and have coordinates (a,b) and (c,d), respectively. What are the coordinates of the vertices B and D? What is the area of the square?

What remainders do you get when square numbers are divided by 4?

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

Find the perimeter and area of a holly leaf that will not lie flat (it has negative curvature with 'circles' having circumference greater than 2πr).

P is a point on the circumference of a circle radius r which rolls, without slipping, inside a circle of radius 2r. What is the locus of P?

Are these statistical statements sometimes, always or never true? Or it is impossible to say?

A circle is inscribed in an equilateral triangle. Smaller circles touch it and the sides of the triangle, the process continuing indefinitely. What is the sum of the areas of all the circles?

Can you work out how to produce the right amount of chemical in a temperature-dependent reaction?

Playing with numbers leads to a conjecture for the solution. Place value and summing a geometric series gives proof of the conjecture.

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You are given the Lowest Common Multiples of sets of digits. Find the digits and then solve the Sudoku.

A game for two people, who take turns to move the counters. The player to remove the last counter from the board wins.