## Most Ridiculous Routes

A postman with time to spare, made a point of finishing his round by walking as far as possible while visiting his last ten houses, which were equally spaced on a straight road.  Assuming a distance of 1 unit between each house, what is the longest distance he could walk, starting at house 1 and visiting each other house once? What if he started at house 2? 3? 4? 5?

Source: Based on The Penguin Book of Curious and Interesting Puzzles, David Wells, #535.

Solutions were received from Alan O'Donnell, Al Zimmermann, Jimmy Chng Gim Hong, Joseph DeVincentis, Jozef Hanenberg, Bernie Erickson, and Saw L.B.  Jozef Hanenberg's solution is a good summary:
Suppose first that the postman returns to his starting point at the end of his round. In that case it doesn't matter where he starts, because his route forms a cycle. To make his route as long as possible he must alternately visit a house in the first half ( 1 to 5 ) and a house in the second half ( 6 to 10 ) Further it doesn't matter in which order he picks his numbers.
The table below illustrates the situation: between two houses is given the number of times that this distance is being walked.

house nr    1        2        3        4        5        6        7        8        9        10
2x        4x      6x     8x      10x     8x      6x      4x        2x

Total distance in such a cycle is 50 units.

If the postman doesn't return to his starting point, he can make his route as long as possible by ending his route at house nr 6, thereby minimizing the number of non-walked units in a cycle-route of 50.
starting point nr 1    45 units
starting point nr 2    46 units
starting point nr 3    47 units
starting point nr 4    48 units
starting point nr 5    49 units.

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