VISITING THE TOWNS.
(
Unicursal and Route Problems)
A traveller, starting from town No. 1, wishes to visit every one of the
towns once, and once only, going only by roads indicated by straight
lines. How many different routes are there from which he can select? Of
course, he must end his journey at No. 1, from which he started, and
must take no notice of cross roads, but go straight from town to town.
This is an absurdly easy puzzle, if you go the right way to work.
Answer:
Note that there are six towns, from which only two roads issue. Thus 1
must lie between 9 and 12 in the circular route. Mark these two roads as
settled. Similarly mark 9, 5, 14, and 4, 8, 14, and 10, 6, 15, and 10,
2, 13, and 3, 7, 13. All these roads must be taken. Then you will find
that he must go from 4 to 15, as 13 is closed, and that he is compelled
to take 3, 11, 16, and also 16, 12. Thus, there is only one route, as
follows: 1, 9, 5, 14, 8, 4, 15, 6, 10, 2, 13, 7, 3, 11, 16, 12, 1, or
its reverse--reading the line the other way. Seven roads are not used.
