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?

Explore the effect of reflecting in two parallel mirror lines.

A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . .

Explore the effect of reflecting in two intersecting mirror lines.

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

Explore the effect of combining enlargements.

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?

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?

What is the volume of the solid formed by rotating this right angled triangle about the hypotenuse?

A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .

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

Consider all two digit numbers (10, 11, . . . ,99). In writing down all these numbers, which digits occur least often, and which occur most often ? What about three digit numbers, four digit numbers. . . .

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. . . .

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?

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.

A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?

Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?

Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

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

Make some loops out of regular hexagons. What rules can you discover?

What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?

If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

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?

Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.

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

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.

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 a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

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

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?

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

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

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?

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

Think of a number, add one, double it, take away 3, add the number you first thought of, add 7, divide by 3 and take away the number you first thought of. You should now be left with 2. How do I. . . .

List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?

Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?

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. . . .

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

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.

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

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. . . .

With one cut a piece of card 16 cm by 9 cm can be made into two pieces which can be rearranged to form a square 12 cm by 12 cm. Explain how this can be done.

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

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

If you have a large supply of 3kg and 8kg weights, how many of each would you need for the average (mean) of the weights to be 6kg?