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.

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

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?

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

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.

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

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?

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

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

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.

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

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.

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?

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?

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.

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.

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

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.

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

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

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. . . .

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

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.

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?

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 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. . . .

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

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

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?

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?

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

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?

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

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

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

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?

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

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?”

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

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

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.

The Egyptians expressed all fractions as the sum of different unit fractions. Here is a chance to explore how they could have written different fractions.

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?

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