Here are the six islands of Transumberg connected by nine bridges.

Can you find a route, starting on any of the islands, that crosses each bridge once?

If you found that puzzle easy try this:

Can you find a route crossing each bridge once (and only once)?

Which town is this?

## A Mathematics Lesson Starter Of The Day

Topics: Starter | Puzzles

• James,
•
• Is this puzzle impossible?
• Mr Rosenberg,
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• Yes, this puzzle is mathematically impossible.
• Transum,
•
• Yes, this impossible puzzle is the famous Seven Bridges of Konigsberg problem which is a well known part of the history of mathematics. Proving that it is impossible leads to graph theory and topology. Do you like puzzles that turn out to be impossible? Some people get very frustrated but miss the point that the positive outcome is the proof that there is no solution. Nowadays Konigsberg is called Kaliningrad and the layout of the town does not look as simple as our diagram above shows. Can you find it on Google Earth and see how many bridges are in place now?

The first puzzle, the six islands of Transumberg, is most definitely possible!
• Matthew, Year 5, Craigslea State School
•
• A very interesting puzzle!
• Connie Bridgeman,
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• It'll make you cross if you can't do it. If you do it you'll cross!
• Enid, Bendigo Violet Street Primary
•
• That's very interesting. I like puzzles that turn out to be impossible, but do get frustrated when I try hundreds of times supposedly for no result. However, working out the different possibilities and discovering the puzzle was impossible was quite a reward!

How did you use this starter? Can you suggest how teachers could present or develop this resource? Do you have any comments? It is always useful to receive feedback and helps make this free resource even more useful for Maths teachers anywhere in the world.

Previous Day | This starter is for 6 August | Next Day

Can you make up your own bridge crossing puzzle with a different number of bridges and islands. How can you predict if the puzzle will be impossible?

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Here is the URL which will take them to a similar type of puzzle.

Transum.org/go/?to=Without

## The Parson's Puzzle

"Here, my worthy Pilgrims, is a strange riddle," quoth the Parson. "Behold how at the branching of the river is an island. Upon this island doth stand my own poor parsonage, and ye may all see the whereabouts of the village church. Mark ye, also, that there be eight bridges and no more over the river in my parish. On my way to church it is my wont to visit sundry of my flock, and in the doing thereof I do pass over every one of the eight bridges once and no more. Can any of ye find the path, after this manner, from the house to the church, without going out of the parish? Nay, nay, my friends, I do never cross the river in any boat, neither by swimming nor wading, nor do I go underground like unto the mole, nor fly in the air as doth the eagle; but only pass over by the[Pg 49] bridges." There is a way in which the Parson might have made this curious journey. Can the reader discover it? At first it seems impossible, but the conditions offer a loophole.

The Canterbury Puzzles, by Henry Ernest Dudeney

## History

In the history of mathematics, the first person go on record with a proven statement about the second problem above was Leonhard Euler in 1736. His 'solution' is considered to be the first theorem of graph theory and the first true proof in the theory of networks, a subject now generally regarded as a branch of combinatorics.

In addition, Euler's recognition that the key information was the number of bridges and the list of their endpoints (rather than their exact positions) is the basis of topology. The difference between the actual layout and the graph schematic is a good example of the idea that topology is not concerned with the rigid shape of objects.

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