If you put three beads onto a tens/ones abacus you could make the numbers 3, 30, 12 or 21. What numbers can be made with six beads?

There are three baskets, a brown one, a red one and a pink one, holding a total of 10 eggs. Can you use the information given to find out how many eggs are in each basket?

Investigate the numbers that come up on a die as you roll it in the direction of north, south, east and west, without going over the path it's already made.

Can you design a new shape for the twenty-eight squares and arrange the numbers in a logical way? What patterns do you notice?

Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?

On a digital clock showing 24 hour time, over a whole day, how many times does a 5 appear? Is it the same number for a 12 hour clock over a whole day?

If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?

Can you find a reliable strategy for choosing coordinates that will locate the robber in the minimum number of guesses?

Imagine you have a large supply of 3kg and 8kg weights. How many of each weight would you need for the average (mean) of the weights to be 6kg? What other averages could you have?

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?

You found some very helpful ways of working on this problem. Some of you drew pictures while others used numbers to represent the circle's dots.

We received good solutions with clear explanations to why the numbers add up to multiples of 11.

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

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

This article gives a brief history of the development of Geometry.

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

Need some help getting started with solving and thinking about rich tasks? Read on for some friendly advice.