1. How to solve Euler paths & circuits. by Mr. King

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

3. Euler Circuits Euler circuits follow these rules: Euler circuits follow these rules: 1. All vertices are equal. 2. You can trace a pencil over every edge without retracing any edges or lifting your pencil, and… 3. You start and finish at the same vertex.

4. Euler Paths Euler paths follow these rules: Euler paths follow these rules: 1. The graph has 2 odd vertices. 2. You can trace a pencil over every edge without retracing or lifting your pencil, but… 3. You will start and finish at different vertices.

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

6. Use a system instead! Instead of scribbling all over your paper, use symbols and make a system. Instead of scribbling all over your paper, use symbols and make a system. Mark your starting point, then use symbols to mark your progress. Mark your starting point, then use symbols to mark your progress. Try using: / X O ~ Try using: / X O ~ Use colors:/ X O ~ Use colors:/ X O ~

7. Use a system instead! Instead of scribbling all over your paper, use symbols and make a system. Instead of scribbling all over your paper, use symbols and make a system. Mark your starting point, then use symbols to mark your progress. Mark your starting point, then use symbols to mark your progress. Try using: / X O ~ Try using: / X O ~ Use colors:/ X O ~ Use colors:/ X O ~

8. Use a system instead! Instead of scribbling all over your paper, use symbols and make a system. Instead of scribbling all over your paper, use symbols and make a system. Mark your starting point, then use symbols to mark your progress. Mark your starting point, then use symbols to mark your progress. Try using: / X O ~ Try using: / X O ~ Use colors:/ X O ~ Use colors:/ X O ~

9. Use a system instead! Instead of scribbling all over your paper, use symbols and make a system. Instead of scribbling all over your paper, use symbols and make a system. Mark your starting point, then use symbols to mark your progress. Mark your starting point, then use symbols to mark your progress. Try using: / X O ~ Try using: / X O ~ Use colors:/ X O ~ Use colors:/ X O ~

10. Use a system instead! Instead of scribbling all over your paper, use symbols and make a system. Instead of scribbling all over your paper, use symbols and make a system. Mark your starting point, then use symbols to mark your progress. Mark your starting point, then use symbols to mark your progress. Try using: / X O ~ Try using: / X O ~ Use colors:/ X O ~ Use colors:/ X O ~

11. Use a system instead! Instead of scribbling all over your paper, use symbols and make a system. Instead of scribbling all over your paper, use symbols and make a system. Mark your starting point, then use symbols to mark your progress. Mark your starting point, then use symbols to mark your progress. Try using: / X O ~ Try using: / X O ~ Use colors:/ X O ~ Use colors:/ X O ~

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

40. Try it yourself- Label your vertices, Label your vertices, Choose a vertex to start at, Choose a vertex to start at, Use a symbol or color to mark your route, Use a symbol or color to mark your route, Write out the route as you go. Write out the route as you go.

41. You can also solve backwards. Check each route by following it. Check each route by following it. Use your symbols to mark the route. Use your symbols to mark the route. Remember the rules for circuits and paths: Remember the rules for circuits and paths: 1.Circuits start and end at the same vertex. 2.Paths start and end at different vertices. 3.You have to go over every edge only once without lifting your pencil.

42. You can also solve backwards. Check each route by following it. Check each route by following it. Use your symbols to mark the route. Use your symbols to mark the route. Remember the rules for circuits and paths: Remember the rules for circuits and paths: 1.Circuits start and end at the same vertex. 2.Paths start and end at different vertices. 3.You have to go over every edge only once without lifting your pencil.

43. You can also solve backwards. Check each route by following it. Check each route by following it. Use your symbols to mark the route. Use your symbols to mark the route. Remember the rules for circuits and paths: Remember the rules for circuits and paths: 1.Circuits start and end at the same vertex. 2.Paths start and end at different vertices. 3.You have to go over every edge only once without lifting your pencil.

44. You can also solve backwards. Check each route by following it. Check each route by following it. Use your symbols to mark the route. Use your symbols to mark the route. Remember the rules for circuits and paths: Remember the rules for circuits and paths: 1.Circuits start and end at the same vertex. 2.Paths start and end at different vertices. 3.You have to go over every edge only once without lifting your pencil.

45. You can also solve backwards. Check each route by following it. Check each route by following it. Use your symbols to mark the route. Use your symbols to mark the route. Remember the rules for circuits and paths: Remember the rules for circuits and paths: 1.Circuits start and end at the same vertex. 2.Paths start and end at different vertices. 3.You have to go over every edge only once without lifting your pencil.

46. You can also solve backwards. Check each route by following it. Check each route by following it. Use your symbols to mark the route. Use your symbols to mark the route. Remember the rules for circuits and paths: Remember the rules for circuits and paths: 1.Circuits start and end at the same vertex. 2.Paths start and end at different vertices. 3.You have to go over every edge only once without lifting your pencil.

47. You can also solve backwards. Check each route by following it. Check each route by following it. Use your symbols to mark the route. Use your symbols to mark the route. Remember the rules for circuits and paths: Remember the rules for circuits and paths: 1.Circuits start and end at the same vertex. 2.Paths start and end at different vertices. 3.You have to go over every edge only once without lifting your pencil.

48. You can also solve backwards. Did you notice what was wrong with route #1? Did you notice what was wrong with route #1? Tell someone sitting next to you why route #1 was neither a Euler path or circuit. Tell someone sitting next to you why route #1 was neither a Euler path or circuit.

49. You can also solve backwards. Route #1 never covered the A-C edge. Route #1 never covered the A-C edge. Let’s try route #2. Let’s try route #2. Don’t forget to use a different symbol or color! Don’t forget to use a different symbol or color!

50. You can also solve backwards. Route #1 never covered the A-C edge. Route #1 never covered the A-C edge. Let’s try route #2. Let’s try route #2.

51. You can also solve backwards. Route #1 never covered the A-C edge. Route #1 never covered the A-C edge. Let’s try route #2. Let’s try route #2.

52. You can also solve backwards. Route #1 never covered the A-C edge. Route #1 never covered the A-C edge. Let’s try route #2. Let’s try route #2.

53. You can also solve backwards. Route #1 never covered the A-C edge. Route #1 never covered the A-C edge. Route #2 doesn’t work. It went over an edge twice. Route #2 doesn’t work. It went over an edge twice.

54. You can also solve backwards. Route #1 never covered the A-C edge. Route #1 never covered the A-C edge. Route #2 doesn’t work. It went over an edge twice. Route #2 doesn’t work. It went over an edge twice. Let’s check #3! Let’s check #3!

55. You can also solve backwards. Route #1 never covered the A-C edge. Route #1 never covered the A-C edge. Route #2 doesn’t work. It went over an edge twice. Route #2 doesn’t work. It went over an edge twice. Let’s check #3! Let’s check #3!

56. You can also solve backwards. Route #1 never covered the A-C edge. Route #1 never covered the A-C edge. Route #2 doesn’t work. It went over an edge twice. Route #2 doesn’t work. It went over an edge twice. Let’s check #3! Let’s check #3!

57. You can also solve backwards. Route #1 never covered the A-C edge. Route #1 never covered the A-C edge. Route #2 doesn’t work. It went over an edge twice. Route #2 doesn’t work. It went over an edge twice. Let’s check #3! Let’s check #3!

58. You can also solve backwards. Route #1 never covered the A-C edge. Route #1 never covered the A-C edge. Route #2 doesn’t work. It went over an edge twice. Route #2 doesn’t work. It went over an edge twice. Let’s check #3! Let’s check #3!

59. You can also solve backwards. Route #1 never covered the A-C edge. Route #1 never covered the A-C edge. Route #2 doesn’t work. It went over an edge twice. Route #2 doesn’t work. It went over an edge twice. Route #3 works! Route #3 works! 1.All edges covered once, 2.Pencil never left paper.

60. Now you try Use symbols Use symbols Mark your route Mark your route Remember rules for paths and circuits. Remember rules for paths and circuits. Eliminate the incorrect options, prove the correct option! Eliminate the incorrect options, prove the correct option!

61. Now you try I hope you chose route #3! I hope you chose route #3!

62. That’s it! Use a system to solve: Use symbols Use symbols Mark your route Mark your route Remember the rules for paths and circuits Remember the rules for paths and circuits Eliminate incorrect options Eliminate incorrect options Prove correct options. Prove correct options.