Search by Topic

Resources tagged with Generalising similar to Multiplication Arithmagons:

Filter by: Content type:
Stage:
Challenge level: Challenge Level:1 Challenge Level:2 Challenge Level:3

There are 122 results

Broad Topics > Using, Applying and Reasoning about Mathematics > Generalising

problem icon

Multiplication Arithmagons

Stage: 4 Challenge Level: Challenge Level:1

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

problem icon

Arithmagons

Stage: 4 Challenge Level: Challenge Level:1

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

problem icon

What's Possible?

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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

problem icon

Polycircles

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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?

problem icon

Gnomon Dimensions

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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

problem icon

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.

problem icon

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.

problem icon

Partitioning Revisited

Stage: 3 Challenge Level: Challenge Level:1

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

problem icon

Thank Your Lucky Stars

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

A counter is placed in the bottom right hand corner of a grid. You toss a coin and move the star according to the following rules: ... What is the probability that you end up in the top left-hand. . . .

problem icon

Loopy

Stage: 4 Challenge Level: Challenge Level:1

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?

problem icon

Pair Products

Stage: 4 Challenge Level: Challenge Level:1

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

problem icon

Adding in Rows

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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?

problem icon

Jam

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

problem icon

More Number Pyramids

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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

problem icon

Nim-like Games

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

A collection of games on the NIM theme

problem icon

Sliding Puzzle

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

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.

problem icon

Make 37

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

Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.

problem icon

Jam

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

A game for 2 players

problem icon

One, Three, Five, Seven

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

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.

problem icon

Number Pyramids

Stage: 3 Challenge Level: Challenge Level:1

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

problem icon

Attractive Tablecloths

Stage: 4 Challenge Level: Challenge Level:1

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?

problem icon

Pentanim

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

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.

problem icon

Nim

Stage: 4 Challenge Level: Challenge Level:1

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 icon

Beelines

Stage: 4 Challenge Level: Challenge Level:1

Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?

problem icon

Nim-interactive

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

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.

problem icon

Handshakes

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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

problem icon

Building Gnomons

Stage: 4 Challenge Level: Challenge Level:1

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

problem icon

Mindreader

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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. . . .

problem icon

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.

problem icon

Special Sums and Products

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level: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.

problem icon

Converging Means

Stage: 3 Challenge Level: Challenge Level:1

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. . . .

problem icon

AMGM

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

problem icon

Enclosing Squares

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

problem icon

Pareq Calc

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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. . . .

problem icon

Lower Bound

Stage: 3 Challenge Level: Challenge Level:1

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

problem icon

Picturing Square Numbers

Stage: 3 Challenge Level: Challenge Level:1

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

problem icon

Sum Equals Product

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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. . . .

problem icon

Painted Cube

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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?

problem icon

More Twisting and Turning

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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

problem icon

Generating Triples

Stage: 4 Challenge Level: Challenge Level:1

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?

problem icon

Partly Painted Cube

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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?

problem icon

Christmas Chocolates

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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

problem icon

Steel Cables

Stage: 4 Challenge Level: Challenge Level:1

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

problem icon

Steps to the Podium

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

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

problem icon

What Numbers Can We Make Now?

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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

problem icon

Route to Infinity

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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

problem icon

Plus Minus

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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

problem icon

Mystic Rose

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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

problem icon

Cubes Within Cubes Revisited

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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?

problem icon

All Tangled Up

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Can you tangle yourself up and reach any fraction?


The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here.
Copyright © 1997 - 2018. University of Cambridge. All rights reserved.
NRICH is part of the family of activities in the Millennium Mathematics Project.