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

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

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

How many moves does it take to swap over some red and blue frogs? Do you have a method?

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.

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

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

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

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?

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?

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

Take a look at the multiplication square. The first eleven triangle numbers have been identified. Can you see a pattern? Does the pattern continue?

A game for 2 players. Set out 16 counters in rows of 1,3,5 and 7. Players take turns to remove any number of counters from a row. The player left with the last counter looses.

Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

A game for 2 players with similarities to NIM. Place one counter on each spot on the games board. Players take it is turns to remove 1 or 2 adjacent counters. The winner picks up the last counter.

A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?

Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?

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?

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

15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?

Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?

Can you find sets of sloping lines that enclose a square?

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

Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

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

In how many ways can you arrange three dice side by side on a surface so that the sum of the numbers on each of the four faces (top, bottom, front and back) is equal?

Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?

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

It starts quite simple but great opportunities for number discoveries and patterns!

An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.

We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4

This task encourages you to investigate the number of edging pieces and panes in different sized windows.

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?

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

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?

Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares?

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?

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

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?”

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

Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.

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.

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.

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

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.

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. . . .