THE GRAND TOUR.
(
Unicursal and Route Problems)
One of the everyday puzzles of life is the working out of routes. If you
are taking a holiday on your bicycle, or a motor tour, there always
arises the question of how you are to make the best of your time and
other resources. You have determined to get as far as some particular
place, to include visits to such-and-such a town, to try to see
something of special interest elsewhere, and perhaps to try to look up
an old friend at a spot that will not take you much out of your way.
Then you have to plan your route so as to avoid bad roads, uninteresting
country, and, if possible, the necessity of a return by the same way
that you went. With a map before you, the interesting puzzle is attacked
and solved. I will present a little poser based on these lines.
I give a rough map of a country--it is not necessary to say what
particular country--the circles representing towns and the dotted lines
the railways connecting them. Now there lived in the town marked A a man
who was born there, and during the whole of his life had never once left
his native place. From his youth upwards he had been very industrious,
sticking incessantly to his trade, and had no desire whatever to roam
abroad. However, on attaining his fiftieth birthday he decided to see
something of his country, and especially to pay a visit to a very old
friend living at the town marked Z. What he proposed was this: that he
would start from his home, enter every town once and only once, and
finish his journey at Z. As he made up his mind to perform this grand
tour by rail only, he found it rather a puzzle to work out his route,
but he at length succeeded in doing so. How did he manage it? Do not
forget that every town has to be visited once, and not more than once.
Answer:
The first thing to do in trying to solve a puzzle like this is to
attempt to simplify it. If you look at Fig. 1, you will see that it is a
simplified version of the map. Imagine the circular towns to be buttons
and the railways to be connecting strings. (See solution to No. 341.)
Then, it will be seen, we have simply "straightened out" the previous
diagram without affecting the conditions. Now we can further simplify by
converting Fig. 1 into Fig. 2, which is a portion of a chessboard. Here
the directions of the railways will resemble the moves of a rook in
chess--that is, we may move in any direction parallel to the sides of
the diagram, but not diagonally. Therefore the first town (or square)
visited must be a black one; the second must be a white; the third must
be a black; and so on. Every odd square visited will thus be black and
every even one white. Now, we have 23 squares to visit (an odd number),
so the last square visited must be black. But Z happens to be white, so
the puzzle would seem to be impossible of solution.
[Illustration: Fig. 1.]
[Illustration: Fig. 2.]
As we were told that the man "succeeded" in carrying put his plan, we
must try to find some loophole in the conditions. He was to "enter every
town once and only once," and we find no prohibition against his
entering once the town A after leaving it, especially as he has never
left it since he was born, and would thus be "entering" it for the first
time in his life. But he must return at once from the first town he
visits, and then he will have only 22 towns to visit, and as 22 is an
even number, there is no reason why he should not end on the white
square Z. A possible route for him is indicated by the dotted line from
A to Z. This route is repeated by the dark lines in Fig. 1, and the
reader will now have no difficulty in applying; it to the original map.
We have thus proved that the puzzle can only be solved by a return to A
immediately after leaving it.
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