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MA111: Contemporary mathematics Jack Schmidt University of Kentucky December 1, 2015 Entrance Slip (due 5 min past the hour): Can you go through all the doors exactly once? Foy Bed Liv Yard Sun Kit Din Write down the path (labelled by

SLIDE 1

MA111: Contemporary mathematics

Jack Schmidt

University of Kentucky

December 1, 2015 Entrance Slip (due 5 min past the hour): Can you go through all the doors exactly once? Bed Sun Liv Kit Foy Din Yard Write down the path (labelled by rooms) or explain why there is no such path. Today we introduce routing problems.

SLIDE 2

Garbage truck route

The garbage truck enters and leaves from the right edge. How should it go through the neighborhood so its claw can get all the trash?

SLIDE 3

The dragon’s tour

When I move to a new place, I like to walk through every doorway It loses some newness if I go through a doorway more than once Can I tour the interior doors of the house without repeats? A B C D E F

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The dragon’s tour with toilets too

When I move to a new place, I need to walk through every doorway I forgot the toilets and closets. Can I tour the interior doors of the house without repeats now? A B C D E F a b e f

SLIDE 5

On patrol again

Can the postal carrier walk every street exactly once They should start and end at the Post Office

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These are all the same question

Mathematics looks at many different problems and finds the common structure A strategy to win all similar games The truck route has roads that connect intersections The house has doors that connect rooms The postal carrier has roads between intersections The generic has edges between vertices we want to travel over all edges

SLIDE 7

The house as a graph

A simpler picture A B C D E F A B C D E F The connections are the same in both pictures

SLIDE 8

The house with toilets as a graph

A simpler picture a b A B C D E F e f A B C D E F a b e f The connections are the same in both pictures

SLIDE 9

The postal patrol

Here is the city as a graph P

SLIDE 10

You draw them

Draw a graph that can be traced Draw a graph that obviously can’t be traced Draw a graph that can’t be traced, but that might take a 5 yr old a little time to figure out Be prepared to draw at the board!

SLIDE 11

Can these be traced?

A B C D Y V X W E Z A B C D E Z W Y V X

SLIDE 12

Can these be traced?

a0 a1 a2 a3 a4 a5 a3 a0 a1 a2 b

SLIDE 13

Assignments and exit slip

Exit slip:

Can you go through all the doors exactly once? Bed Sun Liv Kit Foy Din Yard Write down the path (labelled by rooms) or explain why there is no such path.