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The 7 Bridges of Königsberg River Pregel

Doodles Try drawing each of the doodles below without removing your pencil from the paper and without retracing a line that you have already drawn. Which ones can be drawn?

LC Doodles Charles Dodgson (1832 – 1898) The doodle shown was a favourite of Lewis Carroll. Lewis Carroll was a pseudonym for Charles Dodgson and he was a professor of mathematics at Oxford. He wrote many famous children’s stories including “Alice’s adventures in Wonderland” and “Through the Looking-Glass and What Alice Found There”. He also invented numerous puzzles and games of a mathematical nature. His books contain many coded references to his love of mathematics and numbers. Although he was primarily a mathematician he is credited with producing the greatest nonsense poem in English Literature - “The Jabberwocky”. This poem appears with others in “Through the Looking Glass”, along with characters such as Humpty Dumpty, Tweedle Dum and Tweedle Dee, and the Red Queen.

Charles Dodgson (1832 – 1898) Alice in Wonderland

Through the Looking Glass and What Alice Found There Charles Dodgson (1832 – 1898)

The doodles below are strongly related to the problem of the 7 Bridges of Königsberg. Once you understand the solution to this problem you will be able to tell whether or not any doodle can be drawn, no matter how complex it looks. Doodles ?

Euler Leonhard Euler, (pronounced “oiler”) was a Swiss mathematician whose power of mental calculation was prodigious. As a teenager his fame quickly spread throughout Europe and at the age of 20 he was summoned by Catherine 1 st of Russia to study at the Academy of Science at St Petersburg. He became Professor of Physics at 23 and Professor of mathematics at 26. At the Academy he contributed to numerous branches of mathematics and other sciences, including work on optics and the study of planetary motion. He made progress towards solving “Fermat’s Last Theorem” and introduced the notation for Pi (  ) and sigma (  ). At 31 he lost the sight in his right eye and later became almost totally blind. He had thirteen children of whom only five survived their infancy. His life’s work of 886 books/papers was still being published 50 years after his death The 7 Bridges of Königsberg

Whilst working at the academy Euler got to hear of the problem of the 7 Bridges of K ö nigsberg. The river Pregel ran through the town and divided it into 4 land areas, one of which was an island in the middle. The river was spanned by seven bridges and the people of the town tried to walk a route around the centre that crossed all seven bridges once and only once. The 7 Bridges of Königsberg Use your worksheet to try and find a route around the centre, crossing all seven bridges, once and only once. You can start your journey from any point in the town and you can finish at any point

The 7 Bridges of Königsberg ? I think we need another bridge Becky. River Pregel

Although the people of the town felt that such a journey was impossible, they couldn’t actually prove it. Euler tackled the problem for them and he decided that he didn’t need to make the long journey to Königsberg. Instead, he abstracted the problem and made a sketch of the situation The 7 Bridges of Königsberg Euler represented the land areas by lettered vertices and the bridges by arcs. The resulting diagram he called a network. A B C D Vertex Arc Network I think you might be right Peter. Euler undertook a study of general networks and was able to determine the conditions under which a network was traceable. That is, the network could be fully traced by passing along all arcs once and only once. This is equivalent to drawing the network on paper without lifting your pencil or going over a line once drawn.

Odd and Even The 7 Bridges of Königsberg A B C D Vertex Arc Network During his investigation Euler defined an even vertex and an odd vertex. An odd vertex has an odd number of arcs connected to it and an even vertex has an even number of arcs connected to it. odd even odd

The 7 Bridges of Königsberg A B C D Vertex Arc Network Euler realised that the number of odd vertices that a network contains is a crucial factor in determining whether or not it is traceable. odd even odd

Networks Starting at any vertex, use a pencil (and rubber), to determine whether or not the networks below are traceable. Use the table provided to complete details on the number of odd and even vertices contained in each network. Use your results to predict the traceability of any network. A B C N o of odd verticesN o of even verticesTraceable Y/N A 22Yes B 40No C D E F G H I J K L M N O D E F G H I J K L M N O

Starting at any vertex, use a pencil (and rubber), to determine whether or not the networks below are traceable. Use the table provided to complete details on the number of odd and even vertices contained in each network. Use your results to predict the traceability of any network. A B C N o of odd verticesN o of even verticesTraceable Y/N A 22Yes B 40No C 04Yes D 42No E 03Yes F 41No G 210Yes H 41No I 60 J 23Yes K 40No L 22Yes M 42No N 04Yes O 80No D E F G H I J K L M N O

A network is traceable when the number of odd vertices it contains is ? A B C N o of odd verticesN o of even verticesTraceable Y/N A 22Yes B 40No C 04Yes D 42No E 03Yes F 41No G 210Yes H 41No I 60 J 23Yes K 40No L 22Yes M 42No N 04Yes O 80No D E F G H I J K L M N O 2 or 0 Euler proved that a network is traceable only when it contains exactly 2 odd vertices or if all its vertices are even. Use Euler’s result and network K to explain the answer the 7 bridges problem. Use network L to explain what would happen if an 8 th bridge was built.

The 8 Bridges of Königsberg We’ve got one. River Pregel 8 Bridges

You should now understand why some of the doodles that you did earlier are traceable and others aren’t. Is the doodle below traceable? Doodles ? This network has only 2 odd vertices so it is traceable. You must start and finish at one of these odd vertices in order to complete it. The Lewis Carroll doodle has all vertices even. There are many different traceable paths but you will always have to start and finish at the same vertex.

Can you see why you need 2 odd vertices or all even vertices to complete a traceable path in a network? 1. As you leave, the start vertex is odd. 2. Each time you enter and leave another vertex you add 2 arcs so it remains even Where to finish? 1. Going back to finish at any even vertex will make it odd and will therefore leave the network with just 2 odd vertices. 2. Returning to the start vertex will leave all network vertices even. ? Any network must fulfil one of these conditions for it to be traceable. 1 The diagram below may help. Suppose that you wanted to walk a traceable path between each of the vertices starting at any vertex. In Königsberg Land area Bridge 4 odd vertices  not traceable. There are only two options available! I’ll start here 2 odd vertices with 8 th bridge 

Euler’s work on the 7 Bridges of Königsberg led to a totally new branch of mathematics called Topology.There are many modern day uses and applications stemming from the work that Euler did. They include: Transport and road networks Routing of a variety of pipelines Design and layout of Microchips Design and layout of supermarkets The 7 Bridges of Königsberg

Poem Some citizens of Königsberg were walking on the strand Beside the river Pregel With its seven bridges spanned. ‘O Euler, come and walk with us’, Those burghers did beseech. ‘We’ll roam the seven bridges o’er, And pass but once by each’. ‘It can’t be done’, thus Euler cried. ‘Here comes the QED. Your islands are but vertices And four have odd degree’. ‘From Königsberg to König’s book So runs the graphic tale And still it grows more colourful In Michigan and Yale. Blanche Descartes (The Expanding Unicurse) The problem was even discussed in poetic terms.

Worksheet 1

Worksheet 2 Starting at any vertex, use a pencil (and rubber), to determine whether or not the networks below are traceable. Use the table provided to complete details on the number of odd and even vertices contained in each network. Use your results to predict the traceability of any network. N o of odd verticesN o of even verticesTraceable Y/N A 22Yes B 40No C D E F G H I J K L M N O A B C D E F G H I J K L M N O

Jabberwocky Twas brillig and the slithy toves did gyre and gimble in the wabe: All mimsy were the borogoves, And the mome raths outgrabe. Beware the Jabberwock my son! The jaws that bite, the claws that catch! Beware the Jubjub bird, and shun The frumious Bandersnatch. He took his vorpal sword in hand: Long time the manxome foe he sought So rested he by the Tumtum tree And stood a while in thought. And, as in uffish thought he stood The Jabberwock with eyes of flame, Came whiffling through the tulgey wood, And burbled as it came. One, two! One two! And through and through The vorpal blade went snicker snack! He left it dead, and with its head He went galumphing back. “And has thou slain the Jabberwock? Come to my arms my Beamish boy! O frabjous day, Callooh! Callay He chortled in his joy. Twas brillig and the slithy toves did gyre and gimble in the wabe: All mimsy were the borogoves, And the mome raths outgrabe. The Jabberwocky Through the looking glass and what Alice found there. (Lewis Carroll 1897)

" That's enough to begin with," Humpty Dumpty interrupted; "there are plenty of hard words there. 'Brillig' means four o'clock in the afternoon-the time when you begin broiling things for dinner." "That'll do very well," said Alice. "And 'slithy'?" "Well, 'slithy' means 'lithe and slimy.' 'Lithe' is the same as 'active.' You see, it's like a portmanteau-there are two meanings packed up in one word." "I see it now," Alice remarked thoughtfully. "And what are 'toves'?" "Well, 'toves' are something like badgers-they're something like lizards-and they're something like corkscrews." "They must be very curious looking creatures." "They are at that," said Humpty Dumpty, "also they make their nests under sundials-also they live on cheese." "And what's to 'gyre' and to 'gimble'?" "To 'gyre' is to go round and round like a gyroscope. To 'gimble' is to make holes like a gimlet. "And 'the wabe' is the grass-plot round a sundial, I suppose?" said Alice, surprised at her own ingenuity. "Of course it is. It's called 'wabe,' you know, because it goes a long way before it, and a long way behind it." "And a long way beyond on each side," Alice added. "Exactly so. Well, then, 'mimsy' is flimsy and miserable (there's another portmanteau for you). And a 'borogove' is a thin, shabby-looking bird with its feathers sticking out all round-something like a live mop." "And then 'mome-raths'?" said Alice. "I'm afraid I'm giving you a great deal of trouble." "Well, a 'rath' is a sort of green pig: but 'mome' I'm not certain about. I think it's short for 'from home'- meaning that they'd lost their way, you know." "And what does 'outgrabe' mean?" "Well, 'outgribing' is something between bellowing and whistling, with a kind of sneeze in the middle; however, you'll hear it done, maybe-down in the wood yonder-and when you've once heard it you'll be quite content. Who's been repeating all that hard stuff to you?" "I read it in a book," said Alice. From Chapter 6, Humpty Dumpty