Routes and Networks - July 2004, All Stages

Problems

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Queue

Stage: 1 Challenge Level: Challenge Level:2 Challenge Level:2

25 students are queuing in a straight line. How many are there between Julia and Jenny?

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The Tall Tower

Stage: 1 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

As you come down the ladders of the Tall Tower you collect useful spells. Which way should you go to collect the most spells?

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Here to There 1 2 3

Stage: 1 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Move from the START to the FINISH by moving across or down to the next square. Can you find a route to make these totals?

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Paw Prints

Stage: 2 Challenge Level: Challenge Level:1

A dog is looking for a good place to bury his bone. Can you work out where he started and ended in each case? What possible routes could he have taken?

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A Numbered Route

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

Can you draw a continuous line through 16 numbers on this grid so that the total of the numbers you pass through is as high as possible?

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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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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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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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Lost in Space

Stage: 4 Challenge Level: Challenge Level:1

How many ways are there to count 1 - 2 - 3 in the array of triangular numbers? What happens with larger arrays? Can you predict for any size array?

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All Tied Up

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

A ribbon runs around a box so that it makes a complete loop with two parallel pieces of ribbon on the top. How long will the ribbon be?

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The Spider and the Fly

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

A spider is sitting in the middle of one of the smallest walls in a room and a fly is resting beside the window. What is the shortest distance the spider would have to crawl to catch the fly?

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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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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.