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Broad Topics > Decision Mathematics and Combinatorics > Networks/Graph Theory

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Limiting Probabilities

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Given probabilities of taking paths in a graph from each node, use matrix multiplication to find the probability of going from one vertex to another in 2 stages, or 3, or 4 or even 100.

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Simply Graphs

Stage: 5 Challenge Level: Challenge Level:1

Look for the common features in these graphs. Which graphs belong together?

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Factors and Multiples Graphs

Stage: 4 and 5 Challenge Level: Challenge Level:1

Explore creating 'factors and multiples' graphs such that no lines joining the numbers cross

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Round-robin Scheduling

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

Think about the mathematics of round robin scheduling.

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The Olympic Torch Tour

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

Imagine you had to plan the tour for the Olympic Torch. Is there an efficient way of choosing the shortest possible route?

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Torus Patterns

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

How many different colours would be needed to colour these different patterns on a torus?

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An Introduction to Computer Programming and Mathematics

Stage: 5

This article explains the concepts involved in scientific mathematical computing. It will be very useful and interesting to anyone interested in computer programming or mathematics.

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Neural Nets

Stage: 5

Find out some of the mathematics behind neural networks.

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The Four Colour Theorem

Stage: 3 and 4

The Four Colour Conjecture was first stated just over 150 years ago, and finally proved conclusively in 1976. It is an outstanding example of how old ideas can be combined with new discoveries. prove. . . .

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Symmetric Tangles

Stage: 4

The tangles created by the twists and turns of the Conway rope trick are surprisingly symmetrical. Here's why!

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Tangles

Stage: 3 and 4

A personal investigation of Conway's Rational Tangles. What were the interesting questions that needed to be asked, and where did they lead?

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Euler's Formula

Stage: 5

Some simple ideas about graph theory with a discussion of a proof of Euler's formula relating the numbers of vertces, edges and faces of a graph.

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Going Places with Mathematicians

Stage: 2 and 3

This article looks at the importance in mathematics of representing places and spaces mathematics. Many famous mathematicians have spent time working on problems that involve moving and mapping. . . .

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Some Circuits in Graph or Network Theory

Stage: 4 and 5

Eulerian and Hamiltonian circuits are defined with some simple examples and a couple of puzzles to illustrate Hamiltonian circuits.

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Sprouts Explained

Stage: 2, 3, 4 and 5

This article invites you to get familiar with a strategic game called "sprouts". The game is simple enough for younger children to understand, and has also provided experienced mathematicians with. . . .

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Dice, Routes and Pathways

Stage: 1, 2 and 3

This article for teachers discusses examples of problems in which there is no obvious method but in which children can be encouraged to think deeply about the context and extend their ability to. . . .

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Maximum Flow

Stage: 5 Challenge Level: Challenge Level:1

Given the graph of a supply network and the maximum capacity for flow in each section find the maximum flow across the network.

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Cube Net

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

How many tours visit each vertex of a cube once and only once? How many return to the starting point?

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Konigsberg Plus

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

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.

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Tourism

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

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.

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Travelling Salesman

Stage: 3 Challenge Level: Challenge Level:1

A Hamiltonian circuit is a continuous path in a graph that passes through each of the vertices exactly once and returns to the start. How many Hamiltonian circuits can you find in these graphs?

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Hamilton's Puzzle

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

I start my journey in Rio de Janeiro and visit all the cities as Hamilton described, passing through Canberra before Madrid, and then returning to Rio. What route could I have taken?

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Königsberg

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

Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?

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Magic W Wrap Up

Stage: 5 Challenge Level: Challenge Level:1

Prove that you cannot form a Magic W with a total of 12 or less or with a with a total of 18 or more.

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Sufficient but Not Necessary: Two Eyes and Seki in Go

Stage: 4 and 5

The game of go has a simple mechanism. This discussion of the principle of two eyes in go has shown that the game does not depend on equally clear-cut concepts.

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Behind the Rules of Go

Stage: 4 and 5

This article explains the use of the idea of connectedness in networks, in two different ways, to bring into focus the basics of the game of Go, namely capture and territory.

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Euler's Formula and Topology

Stage: 5

Here is a proof of Euler's formula in the plane and on a sphere together with projects to explore cases of the formula for a polygon with holes, for the torus and other solids with holes and the. . . .

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Classifying Solids Using Angle Deficiency

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

Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry

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The Use of Mathematics in Computer Games

Stage: 5

An account of how mathematics is used in computer games including geometry, vectors, transformations, 3D graphics, graph theory and simulations.

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Ding Dong Bell

Stage: 3, 4 and 5

The reader is invited to investigate changes (or permutations) in the ringing of church bells, illustrated by braid diagrams showing the order in which the bells are rung.

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Geometry and Gravity 2

Stage: 3, 4 and 5

This is the second of two articles and discusses problems relating to the curvature of space, shortest distances on surfaces, triangulations of surfaces and representation by graphs.

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Networks and Nodes

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

Without taking your pencil off the paper or going over a line or passing through one of the points twice, can you follow each of the networks?

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Rail Network

Stage: 2 Challenge Level: Challenge Level:1

This drawing shows the train track joining the Train Yard to all the stations labelled from A to S. Find a way for a train to call at all the stations and return to the Train Yard.

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Diagonal Trace

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

You can trace over all of the diagonals of a pentagon without lifting your pencil and without going over any more than once. Can the same thing be done with a hexagon or with a heptagon?

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Only Connect

Stage: 3 Challenge Level: Challenge Level:1

The graph represents a salesman’s area of activity with the shops that the salesman must visit each day. What route around the shops has the minimum total distance?

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Pattern of Islands

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

In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...

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Redblue

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

Investigate the number of paths you can take from one vertex to another in these 3D shapes. Is it possible to take an odd number and an even number of paths to the same vertex?

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Fermat's Poser

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

Find the point whose sum of distances from the vertices (corners) of a given triangle is a minimum.

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W Mates

Stage: 5 Challenge Level: Challenge Level:1

Show there are exactly 12 magic labellings of the Magic W using the numbers 1 to 9. Prove that for every labelling with a magic total T there is a corresponding labelling with a magic total 30-T.

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Knight Defeated

Stage: 4 Challenge Level: Challenge Level:1

The knight's move on a chess board is 2 steps in one direction and one step in the other direction. Prove that a knight cannot visit every square on the board once and only (a tour) on a 2 by n board. . . .

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Magic W

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

Find all the ways of placing the numbers 1 to 9 on a W shape, with 3 numbers on each leg, so that each set of 3 numbers has the same total.

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Olympic Magic

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

in how many ways can you place the numbers 1, 2, 3 … 9 in the nine regions of the Olympic Emblem (5 overlapping circles) so that the amount in each ring is the same?

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Plum Tree

Stage: 4 and 5 Challenge Level: Challenge Level:1

Label this plum tree graph to make it totally magic!

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Magic Caterpillars

Stage: 4 and 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Label the joints and legs of these graph theory caterpillars so that the vertex sums are all equal.

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Tree Graphs

Stage: 4 Challenge Level: Challenge Level:1

A connected graph is a graph in which we can get from any vertex to any other by travelling along the edges. A tree is a connected graph with no closed circuits (or loops. Prove that every tree. . . .

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Instant Insanity

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

Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.

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Delia's Routes

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

A little mouse called Delia lives in a hole in the bottom of a tree.....How many days will it be before Delia has to take the same route again?