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?”

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 =

For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?

The incircles of 3, 4, 5 and of 5, 12, 13 right angled triangles have radii 1 and 2 units respectively. What about triangles with an inradius of 3, 4 or 5 or ...?

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

What is the value of the integers a and b where sqrt(8-4sqrt3) = sqrt a - sqrt b?

Given a set of points (x,y) with distinct x values, find a polynomial that goes through all of them, then prove some results about the existence and uniqueness of these polynomials.

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.

Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal cuts through any point inside the square?

What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?

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?

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

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

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?

What is the total number of squares that can be made on a 5 by 5 geoboard?

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. . . .

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

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. . . .

Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.

If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.

Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.

Can you find the values at the vertices when you know the values on the edges?

When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?

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?

Can you explain the surprising results Jo found when she calculated the difference between square numbers?

Can you find the area of a parallelogram defined by two vectors?

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?

Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?

ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.

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?

Given that a, b and c are natural numbers show that if sqrt a+sqrt b is rational then it is a natural number. Extend this to 3 variables.

The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?

If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?

Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The loser is the player who takes the last counter.

Here explore some ideas of how the definitions and methods of calculus change if you integrate or differentiate n times when n is not a whole number.

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

Make and prove a conjecture about the cyclic quadrilateral inscribed in a circle of radius r that has the maximum perimeter and the maximum area.

It would be nice to have a strategy for disentangling any tangled ropes...

Charlie has moved between countries and the average income of both has increased. How can this be so?

What's the largest volume of box you can make from a square of paper?

The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.

Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.

Can you work out the irrational numbers that belong in the circles to make the multiplication arithmagon correct?