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

These gnomons appear to have more than a passing connection with the Fibonacci sequence. This problem ask you to investigate some of these connections.

Investigate sequences given by $a_n = \frac{1+a_{n-1}}{a_{n-2}}$ for different choices of the first two terms. Make a conjecture about the behaviour of these sequences. Can you prove your conjecture?

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.

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?

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 winner is the player to take the last counter.

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

To avoid losing think of another very well known game where the patterns of play are similar.

A game for 2 players with similaritlies 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.

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

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.

If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.

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

This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.

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

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

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?

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

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?

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.

Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.

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?

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

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?

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

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?

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

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

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.

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

Can you describe this route to infinity? Where will the arrows take you next?

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?

Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . .

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

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. . . .

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. . . .

Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?

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

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

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?

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

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

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

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?