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#### Resources tagged with Generalising similar to Weekly Problem 11 - 2008:

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### 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?”

### Pair Products

##### Stage: 4 Challenge Level:

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

### Janine's Conjecture

##### Stage: 4 Challenge Level:

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. . . .

### 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?

### 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. . . .

##### Stage: 3 Challenge Level:

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. . . .

### More Number Pyramids

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

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

### AMGM

##### Stage: 4 Challenge Level:

Choose any two numbers. Call them a and b. Work out the arithmetic mean and the geometric mean. Which is bigger? Repeat for other pairs of numbers. What do you notice?

### Chocolate 2010

##### Stage: 4 Challenge Level:

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

### Take Three from Five

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

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

### Multiplication Square

##### Stage: 3 Challenge Level:

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

### 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?

### 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?

### 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.

### Arithmagons

##### Stage: 3 Challenge Level:

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

### AP Rectangles

##### Stage: 3 Challenge Level:

An AP rectangle is one whose area is numerically equal to its perimeter. If you are given the length of a side can you always find an AP rectangle with one side the given length?

### 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?

### 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?

##### 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?

### 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. . . .

### Number Pyramids

##### Stage: 3 Challenge Level:

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

### Tower of Hanoi

##### Stage: 4 Challenge Level:

The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.

##### Stage: 3 Challenge Level:

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. . . .

### 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

### 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?

### Areas of Parallelograms

##### Stage: 4 Challenge Level:

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

### Pentagon

##### Stage: 4 Challenge Level:

Find the vertices of a pentagon given the midpoints of its sides.

### 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?

### Square Pizza

##### Stage: 4 Challenge Level:

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?

### Plus Minus

##### Stage: 4 Challenge Level:

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

### 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?

### Chocolate Maths

##### Stage: 3 Challenge Level:

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. . . .

### 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?

### Partially Painted Cube

##### Stage: 4 Challenge Level:

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?

### Harmonic Triangle

##### Stage: 3 Challenge Level:

Can you see how to build a harmonic triangle? Can you work out the next two rows?

### What Numbers Can We Make Now?

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

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

### Magic Squares II

##### Stage: 4 and 5

An article which gives an account of some properties of magic squares.

### Tourism

##### Stage: 3 Challenge Level:

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.

### Jam

##### Stage: 4 Challenge Level:

A game for 2 players

### 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.

### Go Forth and Generalise

##### Stage: 3

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.

### Winning Lines

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

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

### Painted Cube

##### Stage: 3 Challenge Level:

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?

### Route to Infinity

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

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

### 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?

### 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?

### Steps to the Podium

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

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

### 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?

### Got It

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

A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.

### Magic Letters

##### Stage: 3 Challenge Level:

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