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#### Resources tagged with Generalising similar to Cyclic Quads:

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### Loopy

##### Stage: 4 Challenge Level:

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?

### Gnomon Dimensions

##### Stage: 4 Challenge Level:

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

### Sliding Puzzle

##### Stage: 1, 2, 3 and 4 Challenge Level:

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.

### Polycircles

##### Stage: 4 Challenge Level:

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?

### Jam

##### Stage: 4 Challenge Level:

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

### Multiplication Arithmagons

##### Stage: 4 Challenge Level:

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

### Winning Lines

##### Stage: 2, 3 and 4

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

### Nim-like Games

##### Stage: 2, 3 and 4 Challenge Level:

A collection of games on the NIM theme

### Nim-interactive

##### Stage: 3 and 4 Challenge Level:

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.

### Nim

##### Stage: 4 Challenge Level:

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.

### Picturing Square Numbers

##### Stage: 3 Challenge Level:

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

### Squaring the Circle and Circling the Square

##### Stage: 4 Challenge Level:

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

### One, Three, Five, Seven

##### Stage: 3 and 4 Challenge Level:

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.

### What's Possible?

##### Stage: 4 Challenge Level:

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

### Games Related to Nim

##### Stage: 1, 2, 3 and 4

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.

### Pentanim

##### Stage: 2, 3 and 4 Challenge Level:

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.

### Jam

##### Stage: 4 Challenge Level:

A game for 2 players

### Building Gnomons

##### Stage: 4 Challenge Level:

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

### Multiplication Square

##### Stage: 3 Challenge Level:

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

### Mini-max

##### Stage: 3 Challenge Level:

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. . . .

### Lower Bound

##### Stage: 3 Challenge Level:

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

### Pareq Calc

##### Stage: 4 Challenge Level:

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. . . .

### Handshakes

##### Stage: 3 Challenge Level:

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

### Picturing Triangle Numbers

##### Stage: 3 Challenge Level:

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

### Mystic Rose

##### Stage: 3 Challenge Level:

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

### Special Sums and Products

##### Stage: 3 Challenge Level:

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.

### Sum Equals Product

##### Stage: 3 Challenge Level:

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. . . .

### Plus Minus

##### Stage: 4 Challenge Level:

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

### Enclosing Squares

##### Stage: 3 Challenge Level:

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

### Steel Cables

##### Stage: 4 Challenge Level:

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

### Christmas Chocolates

##### Stage: 3 Challenge Level:

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

### Generating Triples

##### Stage: 4 Challenge Level:

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?

### Three Times Seven

##### Stage: 3 Challenge Level:

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

### Semi-square

##### Stage: 4 Challenge Level:

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?

### Egyptian Fractions

##### Stage: 3 Challenge Level:

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.

##### Stage: 3 Challenge Level:

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?

### AMGM

##### Stage: 4 Challenge Level:

Can you use the diagram to prove the AM-GM inequality?

### Cuboid Challenge

##### Stage: 3 Challenge Level:

What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?

### Sums of Pairs

##### Stage: 3 and 4 Challenge Level:

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?”

### Partitioning Revisited

##### Stage: 3 Challenge Level:

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

### For Richer for Poorer

##### Stage: 3 Challenge Level:

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

### Seven Squares - Group-worthy Task

##### Stage: 3 Challenge Level:

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?

### All Tangled Up

##### Stage: 3 Challenge Level:

Can you tangle yourself up and reach any fraction?

### More Number Pyramids

##### Stage: 3 and 4 Challenge Level:

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

### More Twisting and Turning

##### Stage: 3 Challenge Level:

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

### Attractive Tablecloths

##### Stage: 4 Challenge Level:

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?

### Konigsberg Plus

##### Stage: 3 Challenge Level:

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.

### Shear Magic

##### Stage: 3 Challenge Level:

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

### Steps to the Podium

##### Stage: 2 and 3 Challenge Level:

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

### Cubes Within Cubes Revisited

##### Stage: 3 Challenge Level:

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?