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Breakfast Bytes: Pigeons, Holes, Bridges and Computers

This exploration delves into the historical and mathematical significance of the Seven Bridges of Königsberg problem, posed by Leonard Euler. Discover how abstraction can simplify complex problems and why no solution exists for crossing each bridge exactly once. Additionally, we introduce the Pigeonhole Principle, illustrating its applications with relatable examples such as elevators and socks. This content emphasizes critical thinking and models in problem-solving while showcasing mathematical concepts achievable by anyone interested in learning.

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Breakfast Bytes: Pigeons, Holes, Bridges and Computers

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  1. Breakfast Bytes:Pigeons, Holes, Bridges and Computers Alan Kaylor Cline November 24, 2009

  2. Bridges

  3. Königsberg

  4. How did you say that?

  5. How did you say that? Königsberg

  6. Did you have to write it with the funny o?

  7. Did you have to write it with the funny o? Nope. You can write this with “oe” instead of “ö ”.

  8. Did you have to write it with the funny o? Nope. You can write this with “oe” instead of “ö ”. Koenigsberg

  9. Did you have to write it with the funny o? Nope. You can write this with “oe” instead of “ö ”. Koenigsberg That looks familiar

  10. Koenig+s+berg Koenig

  11. Koenig+s+berg Koenig Don’t we have a street in Austin with that name?

  12. Koenig+s+berg Koenig Don’t we have a street in Austin with that name?

  13. Koenig+s+berg Koenig Don’t we have a street in Austin with that name? Yes, and another way of pronouncing “oe” in German is long A.

  14. So, we could pronounce: Koenig as KAY-nig?

  15. So, we could pronounce: Koenig as KAY-nig? YUP

  16. Does this have anything to do with computer science?

  17. Does this have anything to do with computer science? NOPE

  18. Back to Königsberg Back to Königsberg

  19. Notice the seven bridges Back to Königsberg

  20. Notice the seven bridges 1 2 3 6 4 5 7

  21. Leonard Euler asked: Can you start someplace and return there having crossed each bridge exactly once. 1 2 3 6 4 5 7

  22. You try it.

  23. Couldn’t do it, could you? 1 2 3 6 4 5 7

  24. Couldn’t do it, could you? Can we make this easier to consider? 1 2 3 6 4 5 7

  25. Let’s introduce some dots for the land areas: 1 2 3 6 4 5 7

  26. Let’s introduce some dots for the land areas: 1 2 3 6 4 5 7

  27. … and some lines for the bridges: 1 2 3 6 4 5 7

  28. … and some lines for the bridges: 1 2 3 6 4 5 7

  29. … and remove the picture: 1 2 3 6 4 5 7

  30. … and remove the picture: 1 3 2 6 4 5 7

  31. Do we gain anything by doing this? 1 3 2 6 4 5 7

  32. Do we gain anything by doing this? Sure. It’s easier to concentrate on what matters. 1 3 2 6 4 5 7

  33. The lesson here is that sometimes it’s better to work with a model of the problem that captures just what’s important and no more. 1 3 2 6 4 5 7

  34. The lesson here is that sometimes it’s better to work with a model of the problem that captures just what’s important and no more. We call this abstraction. 1 3 2 6 4 5 7

  35. Can you see from the abstract model why there is no solution to the Seven Bridges of Königsberg Problem? 1 3 2 6 4 5 7

  36. This is a general truth: In any graph, there is a circuit containing every edge exactly once if and only if every node touches an even number of edges.

  37. In the Seven Bridges of Königsberg Problem all four of the nodes have an odd number of edges touching them. 1 3 3 2 5 3 6 4 5 3 7

  38. In the Seven Bridges of Königsberg Problem all four of the nodes have an odd number of edges touching them. So there is no solution. 1 3 3 2 5 3 6 4 5 3 7

  39. In the Seven Bridges of Königsberg Problem all four of the nodes have an odd number of edges touching them. So there is no solution. 1 3 2 6 4 5 7

  40. Now you try this one

  41. Pigeons and Holes

  42. The Pigeonhole Principle

  43. The Pigeonhole Principle “If you have more pigeons than pigeonholes, you can’t stuff the pigeons into the holes without having at least two pigeons in the same hole.”

  44. The Pigeonhole Principle Example 1: Twelve people are on an elevator and they exit on ten different floors. At least two got off on the same floor.

  45. The Pigeonhole Principle Example 2: My house is burning down, it’s dark, and I need a pair of matched socks. If I have socks of 4 colors, how many socks must I take from the drawer to be certain I have a matched pair.

  46. The Pigeonhole Principle Example 3: Let’s everybody select a favorite number between 1 and _.

  47. The Pigeonhole Principle Example 4: Please divide 5 by 7 and obtain at least 13 decimal places.

  48. Let’s divide 5 by 7 and see what happens. .7 7|5.0 4 9 1

  49. Let’s divide 5 by 7 and see what happens. .71 7|5.00 4 9 10 7 3

  50. Let’s divide 5 by 7 and see what happens. .714 7|5.000 4 9 10 7 30 28 2

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