Tourism

2326 /www/nrich/html/content/id/2326/ 1 2011-02-01T00:00:01 <mdoxml version="1.0"> If you can copy a network <ul> <li> without lifting your pen off the paper </li> <li> without drawing any line twice </li> </ul> then it is traversable. Decide which of these diagrams are traversable. <mdo:image width="450" src="tourism.gif" align="top" alt="diagrams with tours and circuits" height="341" bgcolor=""></mdo:image> If you end up where you started when you draw a traversable diagram it is called a traversing circuit . If you don't end up where you started when you draw a traversable diagram it is called a traversing path. Can you give a set of criteria for determining whether a diagram is traversable by a path or a circuit or neither? </mdoxml> <mdoxml version="1.0"> We received a number of good responses to this problem. Eva from Benenden School and Georgina both worked out that the traversable diagrams are: 1, 2, 3, 7, 8, 9, 11. An anonymous contributor explained that: "The rule to find whether a network is traversable or not is by looking at points called nodes. Nodes are places where two or more lines meet. On these networks, the nodes are clearly shown by the black points in the diagrams. Now you are probably wondering what this has to do with the network being traversable or not. The node either would have an odd or even number of lines connected to it. Do not count the nodes with an even number of lines connected to it. Count the number of nodes with an odd number of lines connected to it. If there are no odd nodes or if there are two odd nodes, that means that the network it traversable. Networks with only two odd nodes are in a traversable path and networks with no odd nodes are in a traversable circuit." Georgina added that: "If a node has an even number of lines coming from it, this means that your pen will be able to both enter and leave that node. If not, the pen must start or finish at the node. As you can start and finish at only twonodes (start at one, finish at the other) any network with more than 2 odd nodes is not traversable. If there are no odd nodes this means that the network can be drawn without taking your pen off the paper, starting and finishing at the same place." Sarah from Colyton (and Andrei from School No. 205 in Bucharest) noticed that: "For Curcuits: Every point must have an even number of lines coming out of it." Jonathan and Thomas from Heversham St Peters Primary School and Amy from Farlingaye High School listed their results as follows: "1 is a traversable path 2 is a traversable circuit 3 is a traversable circuit 4 is not traversable 5 is not traversable 6 is not traversable 7 is a traversable circuit 8 is a traversable path 9 is a traversable circuit 10 is not traversable 11 is a traversable path" Yanqing from LCC School summarised her findings as follows: "Look at the nodes: if the node has an even number of lines coming from it, we call it an even node; if it has an odd number of lines, call it an odd node. If the network has NO odd nodes (all even nodes), it is a traversable circuit; if the network has 2 odd nodes, it is a traversable path; if the network has more than 2 odd nodes, it is not traversable." Thank you all for your solutions. </mdoxml> <mdoxml version="1.0"> A good opportunity to develop geometric reasoning, test ideas and formulate hypotheses and generalisations. </mdoxml> <mdoxml version="1.0"> Have a look at the nodes (the points where the lines meet). </mdoxml> <mdoxml version="1.0"> The following diagrams are traversable: 1, 2, 3, 7, 8, 9 and 11 If all the points (nodes) have an even number of paths (arcs) leading from them, the diagram will be a circuit (you can end up where you started): diagrams 2, 3, 7 and 9. If two points (nodes) have an odd number of paths (arcs) leading from them, the diagram will be a tour (you need to start at one of the "odd nodes" and you will finish at the other "odd node"): diagrams 1, 8 and 11. If more than two points (nodes) have an odd number of paths (arcs) leading from them, the diagram will not be traversable: diagrams 4, 5, 6 and 10. NB. You cannot have a diagram with an odd number of odd nodes. </mdoxml> 2 5 0 0 1 0 0 Tourism If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable. Decision Mathematics and Combinatorics Networks/Graph Theory Decision Mathematics and Combinatorics Topology Using, Applying and Reasoning about Mathematics Mathematical reasoning & proof Using, Applying and Reasoning about Mathematics Visualising Using, Applying and Reasoning about Mathematics Generalising