MATHEMATICS 100 Name
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Consider the following map of European cities, and a table of distances (in kilometers) between each pair of cities. Berlin Brussels Düseldorf Luxembourg Münich Brussels 783 Düseldorf 564 223 Luxembourg 764 219 224 Münich 585 771 613 517 Paris 1,057 308 497 375 832 Berlin Brussels Düseldorf Paris Luxembourg Münich Suppose a traveler in Europe wants to visit each of the cities shown on the map exactly once, and the traveler will begin and end his journey in Brussels. How many possible Hamiltonian circuits (you only need to count those which begin and end in Brussels) are there? Explain your answer. Note: you should also take into account order. That is, do you think the Hamiltonian circuit: (a) Brussels Berlin Münich Paris Luxembourg Düseldorf Brussels is the same, or different, than: (b) Brussels Düseldorf Luxembourg Paris Münich Berlin Brussels Just make sure your explanation is consistent with your numeric answer. Starting and stopping in Brussels gives us 5 choices for the next city, followed by 4, then 3, and so on. 5*4*3*2*1 = 120 But, if you consider that the two Hamiltonian circuits above, (a) and (b), are equivalent, then you’d need to divide this by 2 to give the total number of unique HC’s, which would be 60. Now, of the Hamiltonian circuit’s starting and stopping in Brussels, give the one which minimizes the total distance traveled. Hints: Use the map to narrow down the possible circuits to a few, then use the table to find the total distance for these remaining possible circuits. Brussels Düseldorf Berlin Münich Luxembourg Paris Brussels 223 + 564 + 585 + 517 + 375 + 308 = 2,572 Brussels Luxembourg Düseldorf Berlin Münich Paris Brussels 219 + 224 + 564 + 585 + 832 + 308 = 2,732 Brussels Düseldorf Berlin Münich Paris Luxembourg Brussels 223 + 564 + 585 + 832 + 375 + 219 = 2,798 Brussels Düseldorf Luxembourg Berlin Münich Paris Brussels 223 + 224 + 764 + 585 + 832 + 308 = 2,936 .