[00:08.63]FINDING THE SOLUTION
[00:12.06]Do you like puzzles? Euler did.
[00:16.55]Did you solve the one you heard for the listening task? No!
[00:21.73]Well,don't worry, Euler didn't either!
[00:24.73]As he loved mathematical puzzles,
[00:27.97]he wanted to know why this one wouldn' t work.
[00:31.22]So he walked around the town and over the bridges of Konigsberg several times.
[00:36.96]To his surprise,he found that he could cross six of the bridges
[00:41.64]without going over any of them twice or going back on himself(see Fig 3),
[00:47.56]but he couldn't cross all seven.
[00:50.44]He just had to know why.
[00:54.18]So he decided to look at the problem another way.
[00:56.55]He drew himself a picture of the town and the seven bridges like the one above.
[01:02.67]He marked the land and the bridges.
[01:05.79]Then he put a dot or point into the centre of each of the areas of land.
[01:11.40]He joined these points together using curved lines to go over the bridges(see Fig 1).
[01:20.33]He noticed that some points had three lines going to them(A,B and C) and one had five(D).
[01:30.43]He wondered if this was important and why the puzzle would not work.
[01:35.36]As three and five are odd numbers he called them"odd" points.
[01:40.79]To make the puzzle clearer he took away the bridges to see the patten more clearly(see Fig 2).
[01:49.47]He wondered whether the puzzle would work if he took one bridge away(as in Fig 3).
[01:55.58]This time the diagram was simpler(as in Fig 4).
[02:01.08]He counted the lines going to points A,B,C and D.
[02:07.38]This time they were different.
[02:10.37]Two of them had even numbers of lines(B had two and D had four).
[02:16.24]Two and four are both even numbers so Euler called them" even" points.
[02:22.41]Two points in Fig 4 had an odd number of lines going to them(A and C both had three) and so he called them"odd" points.
[02:36.33]Using this new diagram Euler started at point A,
[02:39.95]went along the straight line to B and then to C.
[02:43.75]Then he followed the curved line through D and back to A.
[02:50.31]Finally he followed the other curved line from A back through D to C where he finished the pattern.
[02:57.64]This time it worked.
[02:59.51]He had been able to go over the figure visiting each point
[03:03.82]but not going over any line twice or lifting his pencil from the page.
[03:09.12]Euler became very excited.
[03:12.68]Now he knew that the number of odd points was the key to the puzzle.
[03:17.23]However,you still needed some even points in your figure if you wanted it to work.
[03:23.47]So Euler looked for a general rule:
[03:26.29]If a figure has more than two odd points,
[03:29.91]you cannot go over it without lifting your pencil from the page or going over a line twice.
[03:36.02]Quickly he went to his textbooks to find some more figures.
[03:41.20]He looked at the four diagrams shown below and found that when he used his rule,
[03:47.06]he could tell if he could go over the whole figure without taking his pencil from the paper.
[03:53.61]He was overjoyed.
[03:54.86]He did not know it but his little puzzle had started a whole new branch of mathematics called"topology".
[04:01.72]In his honour this puzzle is called" finding the Euler path".