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Resources tagged with Generalising similar to Building Gnomons:

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Building Gnomons

Stage: 4 Challenge Level:

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

Jam

Stage: 4 Challenge Level:

A game for 2 players

Hypotenuse Lattice Points

Stage: 4 Challenge Level:

The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN?

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.

Jam

Stage: 4 Challenge Level:

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

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?

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.

Route to Infinity

Stage: 3 and 4 Challenge Level:

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

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.

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.

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.

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.

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.

Problem Solving, Using and Applying and Functional Mathematics

Stage: 1, 2, 3, 4 and 5 Challenge Level:

Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.

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.

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?

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

Multiplication Square

Stage: 3 Challenge Level:

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

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?

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?

Pentagon

Stage: 4 Challenge Level:

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

Number Pyramids

Stage: 3 Challenge Level:

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

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?

Of All the Areas

Stage: 4 Challenge Level:

Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?

Areas of Parallelograms

Stage: 4 Challenge Level:

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

Equilateral Areas

Stage: 4 Challenge Level:

ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.

Happy Numbers

Stage: 3 Challenge Level:

Take any whole number between 1 and 999, add the squares of the digits to get a new number. Make some conjectures about what happens in general.

Summing Consecutive Numbers

Stage: 3 Challenge Level:

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

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?

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?

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?

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?

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?”

In a Spin

Stage: 4 Challenge Level:

What is the volume of the solid formed by rotating this right angled triangle about the hypotenuse?

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?

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.

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?

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...

Christmas Chocolates

Stage: 3 Challenge Level:

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

More Magic Potting Sheds

Stage: 3 Challenge Level:

The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

Nim-7

Stage: 1, 2 and 3 Challenge Level:

Can you work out how to win this game of Nim? Does it matter if you go first or second?

Magic Squares

Stage: 4 and 5

An account of some magic squares and their properties and and how to construct them for yourself.

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?

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. . . .

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?

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. . . .

Magic Squares II

Stage: 4 and 5

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

Plus Minus

Stage: 4 Challenge Level:

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

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?