How to solve Euler How to solve Euler paths & circuits.paths & circuits.

by Mr. Kingby Mr. King

First – let’s review…First – let’s review…

What is a Euler circuit?What is a Euler circuit? What is a Euler path?What is a Euler path?

Tell someone next to you what you Tell someone next to you what you remember.remember.

Euler CircuitsEuler Circuits Euler circuits Euler circuits

follow these rules:follow these rules:1.1. All vertices are All vertices are

equalequal..2.2. You can trace a You can trace a

pencil over every pencil over every edge without edge without retracing any edges retracing any edges or lifting your pencil, or lifting your pencil, and…and…

3.3. You You start and finish start and finish at the same at the same vertexvertex..

Euler PathsEuler Paths Euler paths follow Euler paths follow

these rules:these rules:1.1. The graph has The graph has 2 odd2 odd

vertices.vertices.2.2. You can trace a You can trace a

pencil over every pencil over every edge without edge without retracing or lifting retracing or lifting your pencil, but…your pencil, but…

3.3. You will You will start and start and finish at different finish at different verticesvertices..

Back to solving…Back to solving…

You probably noticed that solving Euler You probably noticed that solving Euler paths and circuits can be messy and paths and circuits can be messy and confusing. When you trace with a pencil, confusing. When you trace with a pencil, you may end up with this-you may end up with this-

Use a system instead!Use a system instead!

Instead of scribbling Instead of scribbling all over your paper, all over your paper, use symbols and make use symbols and make a system.a system.

Mark your starting Mark your starting point, then use point, then use symbols to mark your symbols to mark your progress.progress.

Try using: Try using: / X O ~/ X O ~ Use colors:Use colors:// XX OO ~~

Use a system instead!Use a system instead!

Instead of scribbling Instead of scribbling all over your paper, all over your paper, use symbols and make use symbols and make a system.a system.

Mark your starting Mark your starting point, then use point, then use symbols to mark your symbols to mark your progress.progress.

Try using: Try using: / X O ~/ X O ~ Use colors:Use colors:// XX OO ~~

Use a system instead!Use a system instead!

Instead of scribbling Instead of scribbling all over your paper, all over your paper, use symbols and make use symbols and make a system.a system.

Mark your starting Mark your starting point, then use point, then use symbols to mark your symbols to mark your progress.progress.

Try using: Try using: / X O ~/ X O ~ Use colors:Use colors:// XX OO ~~

Use a system instead!Use a system instead!

Instead of scribbling Instead of scribbling all over your paper, all over your paper, use symbols and make use symbols and make a system.a system.

Mark your starting Mark your starting point, then use point, then use symbols to mark your symbols to mark your progress.progress.

Try using: Try using: / X O ~/ X O ~ Use colors:Use colors:// XX OO ~~

Use a system instead!Use a system instead!

Instead of scribbling Instead of scribbling all over your paper, all over your paper, use symbols and make use symbols and make a system.a system.

Mark your starting Mark your starting point, then use point, then use symbols to mark your symbols to mark your progress.progress.

Try using: Try using: / X O ~/ X O ~ Use colors:Use colors:// XX OO ~~

Use a system instead!Use a system instead!

Instead of scribbling Instead of scribbling all over your paper, all over your paper, use symbols and make use symbols and make a system.a system.

Mark your starting Mark your starting point, then use point, then use symbols to mark your symbols to mark your progress.progress.

Try using: Try using: / X O ~/ X O ~ Use colors:Use colors:// XX OO ~~

Switch symbols when you fail, and Switch symbols when you fail, and label your vertices to mark your label your vertices to mark your

progressprogress

Switch symbols when you fail, and Switch symbols when you fail, and label your vertices to mark your label your vertices to mark your

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Switch symbols when you fail, and Switch symbols when you fail, and label your vertices to mark your label your vertices to mark your

progressprogress

Switch symbols when you fail, and Switch symbols when you fail, and label your vertices to mark your label your vertices to mark your

progressprogress

Switch symbols when you fail, and Switch symbols when you fail, and label your vertices to mark your label your vertices to mark your

progressprogress

Switch symbols when you fail, and Switch symbols when you fail, and label your vertices to mark your label your vertices to mark your

progressprogress

Switch symbols when you fail, and Switch symbols when you fail, and label your vertices to mark your label your vertices to mark your

progressprogress

Switch symbols when you fail, and Switch symbols when you fail, and label your vertices to mark your label your vertices to mark your

progressprogress

Switch symbols when you fail, and Switch symbols when you fail, and label your vertices to mark your label your vertices to mark your

progressprogress

Switch symbols when you fail, and Switch symbols when you fail, and label your vertices to mark your label your vertices to mark your

progressprogress

Switch symbols when you fail, and Switch symbols when you fail, and label your vertices to mark your label your vertices to mark your

progressprogress

Switch symbols when you fail, and Switch symbols when you fail, and label your vertices to mark your label your vertices to mark your

progressprogress

Switch symbols when you fail, and Switch symbols when you fail, and label your vertices to mark your label your vertices to mark your

progressprogress

Switch symbols when you fail, and Switch symbols when you fail, and label your vertices to mark your label your vertices to mark your

progressprogress

Switch symbols when you fail, and Switch symbols when you fail, and label your vertices to mark your label your vertices to mark your

progressprogress

Switch symbols when you fail, and Switch symbols when you fail, and label your vertices to mark your label your vertices to mark your

progressprogress

Switch symbols when you fail, and Switch symbols when you fail, and label your vertices to mark your label your vertices to mark your

progressprogress

Switch symbols when you fail, and Switch symbols when you fail, and label your vertices to mark your label your vertices to mark your

progressprogress

Switch symbols when you fail, and Switch symbols when you fail, and label your vertices to mark your label your vertices to mark your

progressprogress

Switch symbols when you fail, and Switch symbols when you fail, and label your vertices to mark your label your vertices to mark your

progressprogress

Switch symbols when you fail, and Switch symbols when you fail, and label your vertices to mark your label your vertices to mark your

progressprogress

Switch symbols when you fail, and Switch symbols when you fail, and label your vertices to mark your label your vertices to mark your

progressprogress

Switch symbols when you fail, and Switch symbols when you fail, and label your vertices to mark your label your vertices to mark your

progressprogress

Switch symbols when you fail, and Switch symbols when you fail, and label your vertices to mark your label your vertices to mark your

progressprogress

Switch symbols when you fail, and Switch symbols when you fail, and label your vertices to mark your label your vertices to mark your

progressprogress

Switch symbols when you fail, and Switch symbols when you fail, and label your vertices to mark your label your vertices to mark your

progressprogress

Switch symbols when you fail, and Switch symbols when you fail, and label your vertices to mark your label your vertices to mark your

progressprogress

Switch symbols when you fail, and Switch symbols when you fail, and label your vertices to mark your label your vertices to mark your

progressprogress

Try it yourself-Try it yourself-

Label your vertices,Label your vertices, Choose a vertex to Choose a vertex to

start at,start at, Use a symbol or Use a symbol or

color to mark your color to mark your route,route,

Write out the route Write out the route as you go.as you go.

You can also solve backwards.You can also solve backwards. Check each route by Check each route by

following it.following it. Use your symbols to Use your symbols to

mark the route.mark the route. Remember the rules Remember the rules

for circuits and for circuits and paths:paths:1.1. Circuits start and end Circuits start and end

at the same vertex.at the same vertex.2.2. Paths start and end at Paths start and end at

different vertices.different vertices.3.3. You have to go over You have to go over

every edge only once every edge only once without lifting your without lifting your pencil.pencil.

You can also solve backwards.You can also solve backwards. Check each route by Check each route by

following it.following it. Use your symbols to Use your symbols to

mark the route.mark the route. Remember the rules Remember the rules

for circuits and for circuits and paths:paths:1.1. Circuits start and end Circuits start and end

at the same vertex.at the same vertex.2.2. Paths start and end at Paths start and end at

different vertices.different vertices.3.3. You have to go over You have to go over

every edge only once every edge only once without lifting your without lifting your pencil.pencil.

You can also solve backwards.You can also solve backwards. Check each route by Check each route by

following it.following it. Use your symbols to Use your symbols to

mark the route.mark the route. Remember the rules Remember the rules

for circuits and for circuits and paths:paths:1.1. Circuits start and end Circuits start and end

at the same vertex.at the same vertex.2.2. Paths start and end at Paths start and end at

different vertices.different vertices.3.3. You have to go over You have to go over

every edge only once every edge only once without lifting your without lifting your pencil.pencil.

You can also solve backwards.You can also solve backwards. Check each route by Check each route by

following it.following it. Use your symbols to Use your symbols to

mark the route.mark the route. Remember the rules Remember the rules

for circuits and for circuits and paths:paths:1.1. Circuits start and end Circuits start and end

at the same vertex.at the same vertex.2.2. Paths start and end at Paths start and end at

different vertices.different vertices.3.3. You have to go over You have to go over

every edge only once every edge only once without lifting your without lifting your pencil.pencil.

You can also solve backwards.You can also solve backwards. Check each route by Check each route by

following it.following it. Use your symbols to Use your symbols to

mark the route.mark the route. Remember the rules Remember the rules

for circuits and for circuits and paths:paths:1.1. Circuits start and end Circuits start and end

at the same vertex.at the same vertex.2.2. Paths start and end at Paths start and end at

different vertices.different vertices.3.3. You have to go over You have to go over

every edge only once every edge only once without lifting your without lifting your pencil.pencil.

You can also solve backwards.You can also solve backwards. Check each route by Check each route by

following it.following it. Use your symbols to Use your symbols to

mark the route.mark the route. Remember the rules Remember the rules

for circuits and for circuits and paths:paths:1.1. Circuits start and end Circuits start and end

at the same vertex.at the same vertex.2.2. Paths start and end at Paths start and end at

different vertices.different vertices.3.3. You have to go over You have to go over

every edge only once every edge only once without lifting your without lifting your pencil.pencil.

You can also solve backwards.You can also solve backwards. Check each route by Check each route by

following it.following it. Use your symbols to Use your symbols to

mark the route.mark the route. Remember the rules Remember the rules

for circuits and for circuits and paths:paths:1.1. Circuits start and end Circuits start and end

at the same vertex.at the same vertex.2.2. Paths start and end at Paths start and end at

different vertices.different vertices.3.3. You have to go over You have to go over

every edge only once every edge only once without lifting your without lifting your pencil.pencil.

You can also solve backwards.You can also solve backwards.

Did you notice Did you notice what was wrong what was wrong with route #1?with route #1?

Tell someone Tell someone sitting next to you sitting next to you why route #1 was why route #1 was neither a Euler neither a Euler path or circuit.path or circuit.

You can also solve backwards.You can also solve backwards.

Route #1 never Route #1 never covered the A-C covered the A-C edge.edge.

Let’s try route Let’s try route #2.#2.

Don’t forget to Don’t forget to use a different use a different symbol or color!symbol or color!

You can also solve backwards.You can also solve backwards.

Route #1 never Route #1 never covered the A-C covered the A-C edge.edge.

Let’s try route #2. Let’s try route #2.

You can also solve backwards.You can also solve backwards.

Route #1 never Route #1 never covered the A-C covered the A-C edge.edge.

Let’s try route #2. Let’s try route #2.

You can also solve backwards.You can also solve backwards.

Route #1 never Route #1 never covered the A-C covered the A-C edge.edge.

Let’s try route #2. Let’s try route #2.

You can also solve backwards.You can also solve backwards.

Route #1 never Route #1 never covered the A-C covered the A-C edge.edge.

Route #2 Route #2 doesn’t work. It doesn’t work. It went over an went over an edge twice.edge twice.

You can also solve backwards.You can also solve backwards.

Route #1 never Route #1 never covered the A-C covered the A-C edge.edge.

Route #2 doesn’t Route #2 doesn’t work. It went work. It went over an edge over an edge twice.twice.

Let’s check #3!Let’s check #3!

You can also solve backwards.You can also solve backwards.

Route #1 never Route #1 never covered the A-C covered the A-C edge.edge.

Route #2 doesn’t Route #2 doesn’t work. It went work. It went over an edge over an edge twice.twice.

Let’s check #3!Let’s check #3!

You can also solve backwards.You can also solve backwards.

Route #1 never Route #1 never covered the A-C covered the A-C edge.edge.

Route #2 doesn’t Route #2 doesn’t work. It went work. It went over an edge over an edge twice.twice.

Let’s check #3!Let’s check #3!

You can also solve backwards.You can also solve backwards.

Route #1 never Route #1 never covered the A-C covered the A-C edge.edge.

Route #2 doesn’t Route #2 doesn’t work. It went work. It went over an edge over an edge twice.twice.

Let’s check #3!Let’s check #3!

You can also solve backwards.You can also solve backwards.

Route #1 never Route #1 never covered the A-C covered the A-C edge.edge.

Route #2 doesn’t Route #2 doesn’t work. It went work. It went over an edge over an edge twice.twice.

Let’s check #3!Let’s check #3!

You can also solve backwards.You can also solve backwards.

Route #1 never Route #1 never covered the A-C covered the A-C edge.edge.

Route #2 doesn’t Route #2 doesn’t work. It went over work. It went over an edge twice.an edge twice.

Route #3 works!Route #3 works!1.1. All edges covered All edges covered

once,once,

2.2. Pencil never left Pencil never left paper.paper.

Now you tryNow you try

Use symbolsUse symbols Mark your routeMark your route Remember rules for Remember rules for

paths and circuits.paths and circuits.

Eliminate the Eliminate the incorrect options, incorrect options, prove the correct prove the correct option!option!

Now you tryNow you try

I hope you I hope you chose route #3!chose route #3!

That’s it!That’s it!

Use a system to solve: Use a system to solve: Use symbolsUse symbols Mark your routeMark your route Remember the rules for paths and Remember the rules for paths and

circuitscircuits Eliminate incorrect optionsEliminate incorrect options Prove correct options.Prove correct options.