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There are **45** NRICH Mathematical resources connected to **Networks/graph theory**, you may find related items under Decision mathematics and combinatorics.

Problem
Primary curriculum
Secondary curriculum
### Limiting Probabilities

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.

Age 16 to 18

Challenge Level

Article
Primary curriculum
Secondary curriculum
### A Curious Collection of Bridges

Read about the problem that tickled Euler's curiosity and led to a new branch of mathematics!

Age 11 to 18

Problem
Primary curriculum
Secondary curriculum
### Simply Graphs

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Factors and Multiples Graphs

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Round-robin Scheduling

Think about the mathematics of round robin scheduling.

Age 7 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### The Olympic Torch Tour

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Torus Patterns

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

Age 16 to 18

Challenge Level

Article
Primary curriculum
Secondary curriculum
### An Introduction to Computer Programming and Mathematics

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.

Age 16 to 18

Article
Primary curriculum
Secondary curriculum
### The Four Colour Theorem

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 a mathematical theorem.

Age 11 to 16

Article
Primary curriculum
Secondary curriculum
### Symmetric Tangles

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

Age 14 to 16

Article
Primary curriculum
Secondary curriculum
### Tangles

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

Age 11 to 16

Article
Primary curriculum
Secondary curriculum
### Euler's Formula

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.

Age 16 to 18

Article
Primary curriculum
Secondary curriculum
### Going Places with Mathematicians

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 things.

Age 7 to 14

Article
Primary curriculum
Secondary curriculum
### Some Circuits in Graph or Network Theory

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

Age 14 to 18

Article
Primary curriculum
Secondary curriculum
### Sprouts Explained

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 significant food for thought.

Age 7 to 18

Article
Primary curriculum
Secondary curriculum
### Dice, Routes and Pathways

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 think mathematically, especially geometrically.

Age 5 to 14

Problem
Primary curriculum
Secondary curriculum
### Maximum Flow

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Cube Net

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### The Bridges of Konigsberg

Investigate how networks can be used to solve a problem for the 18th Century inhabitants of Konigsberg.

Age 11 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Tourism

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.

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Travelling Salesman

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?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Hamilton's Puzzle

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?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### KÃ¶nigsberg

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?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Magic W Wrap Up

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.

Age 16 to 18

Challenge Level

Article
Primary curriculum
Secondary curriculum
### Sufficient but Not Necessary: Two Eyes and Seki in Go

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.

Age 14 to 18

Article
Primary curriculum
Secondary curriculum
### Behind the Rules of Go

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.

Age 14 to 18

Article
Primary curriculum
Secondary curriculum
### Euler's Formula and Topology

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 relationship between Euler's formula and angle deficiency of polyhedra.

Age 16 to 18

Problem
Primary curriculum
Secondary curriculum
### Classifying Solids Using Angle Deficiency

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

Age 11 to 16

Challenge Level

Article
Primary curriculum
Secondary curriculum
### The Use of Mathematics in Computer Games

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

Age 16 to 18

Article
Primary curriculum
Secondary curriculum
### Ding Dong Bell

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.

Age 11 to 18

Article
Primary curriculum
Secondary curriculum
### Geometry and Gravity 2

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.

Age 11 to 18

Problem
Primary curriculum
Secondary curriculum
### Networks and Nodes

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?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Only Connect

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?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Pattern of Islands

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

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Redblue

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?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Fermat's Poser

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### W Mates

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.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Knight Defeated

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 for any value of n. How many ways can a knight do this on a 3 by 4 board?

Age 14 to 16

Challenge Level

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Primary curriculum
Secondary curriculum
### Magic W

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.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Olympic Magic

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?

Age 14 to 16

Challenge Level

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Primary curriculum
Secondary curriculum
### Magic Caterpillars

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

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Network Trees

Explore some of the different types of network, and prove a result about network trees.

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Instant Insanity

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.

Age 11 to 18

Challenge Level