A, B & C own a half, a third and a sixth of a coin collection. Each grab some coins, return some, then share equally what they had put back, finishing with their own share. How rich are they?

Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?

A mother wants to share a sum of money by giving each of her children in turn a lump sum plus a fraction of the remainder. How can she do this in order to share the money out equally?

Show that all pentagonal numbers are one third of a triangular number.

How many more miles must the car travel before the numbers on the milometer and the trip meter contain the same digits in the same order?

A cyclist and a runner start off simultaneously around a race track each going at a constant speed. The cyclist goes all the way around and then catches up with the runner. He then instantly turns. . . .

Can you find a rule which connects consecutive triangular numbers?

A job needs three men but in fact six people do it. When it is finished they are all paid the same. How much was paid in total, and much does each man get if the money is shared as Fred suggests?

First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...

Think of a number... follow the machine's instructions. I know what your number is! Can you explain how I know?

Think of a number and follow my instructions. Tell me your answer, and I'll tell you what you started with! Can you explain how I know?

Take a few whole numbers away from a triangle number. If you know the mean of the remaining numbers can you find the triangle number and which numbers were removed?

Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”

The number 27 is special because it is three times the sum of its digits 27 = 3 (2 + 7). Find some two digit numbers that are SEVEN times the sum of their digits (seven-up numbers)?

A box has faces with areas 3, 12 and 25 square centimetres. What is the volume of the box?

Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?

Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?

The well known Fibonacci sequence is 1 ,1, 2, 3, 5, 8, 13, 21.... How many Fibonacci sequences can you find containing the number 196 as one of the terms?

Use algebra to reason why 16 and 32 are impossible to create as the sum of consecutive numbers.

Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400. . . .

Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .

Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .

Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

My train left London between 6 a.m. and 7 a.m. and arrived in Paris between 9 a.m. and 10 a.m. At the start and end of the journey the hands on my watch were in exactly the same positions but the. . . .

Semicircles are drawn on the sides of a rectangle ABCD. A circle passing through points ABCD carves out four crescent-shaped regions. Prove that the sum of the areas of the four crescents is equal to. . . .

Two semi-circles (each of radius 1/2) touch each other, and a semi-circle of radius 1 touches both of them. Find the radius of the circle which touches all three semi-circles.

If the sides of the triangle in the diagram are 3, 4 and 5, what is the area of the shaded square?

Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.

A moveable screen slides along a mirrored corridor towards a centrally placed light source. A ray of light from that source is directed towards a wall of the corridor, which it strikes at 45 degrees. . . .

Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?

How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?

Think of a two digit number, reverse the digits, and add the numbers together. Something special happens...

How many winning lines can you make in a three-dimensional version of noughts and crosses?

Can you make sense of these three proofs of Pythagoras' Theorem?

Crosses can be drawn on number grids of various sizes. What do you notice when you add opposite ends?

Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

The sums of the squares of three related numbers is also a perfect square - can you explain why?

How good are you at finding the formula for a number pattern ?

Attach weights of 1, 2, 4, and 8 units to the four attachment points on the bar. Move the bar from side to side until you find a balance point. Is it possible to predict that position?

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?

32 x 38 = 30 x 40 + 2 x 8; 34 x 36 = 30 x 40 + 4 x 6; 56 x 54 = 50 x 60 + 6 x 4; 73 x 77 = 70 x 80 + 3 x 7 Verify and generalise if possible.

The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for. . . .

Fifteen students had to travel 60 miles. They could use a car, which could only carry 5 students. As the car left with the first 5 (at 40 miles per hour), the remaining 10 commenced hiking along the. . . .