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?

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?

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?

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?

What are the possible dimensions of a rectangular hallway if the number of tiles around the perimeter is exactly half the total number of tiles?

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

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

There is a particular value of x, and a value of y to go with it, which make all five expressions equal in value, can you find that x, y pair ?

Create some shapes by combining two or more rectangles. What can you say about the areas and perimeters of the shapes you can make?

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

Can you find the lap times of the two cyclists travelling at constant speeds?

Brian swims at twice the speed that a river is flowing, downstream from one moored boat to another and back again, taking 12 minutes altogether. How long would it have taken him in still water?

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.

Can you see how to build a harmonic triangle? Can you work out the next two rows?

Is there a temperature at which Celsius and Fahrenheit readings are the same?

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

A task which depends on members of the group noticing the needs of others and responding.

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?

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?

If you know the perimeter of a right angled triangle, what can you say about the area?

The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?

Play around with the Fibonacci sequence and discover some surprising results!

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. . . .

Account of an investigation which starts from the area of an annulus and leads to the formula for the difference of two squares.

What is special about the difference between squares of numbers adjacent to multiples of three?

Can you explain what is going on in these puzzling number tricks?

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)?

There are unexpected discoveries to be made about square numbers...

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. . . .

Robert noticed some interesting patterns when he highlighted square numbers in a spreadsheet. Can you prove that the patterns will continue?

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. . . .

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. . . .

The heptathlon is an athletics competition consisting of 7 events. Can you make sense of the scoring system in order to advise a heptathlete on the best way to reach her target?

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. . . .

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

The Number Jumbler can always work out your chosen symbol. Can you work out how?

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.

115^2 = (110 x 120) + 25, that is 13225 895^2 = (890 x 900) + 25, that is 801025 Can you explain what is happening and generalise?

Find the five distinct digits N, R, I, C and H in the following nomogram

Where should you start, if you want to finish back where you started?

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?

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

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?

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

If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.

Can you figure out how sequences of beach huts are generated?