248.—­THE CYCLIST’S TOUR.

When Mr. Maggs replied, “No way, I’m sure,” he was not saying that the thing was impossible, but was really giving the actual route by which the problem can be solved.  Starting from the star, if you visit the towns in the order, NO WAY, I’M SURE, you will visit every town once, and only once, and end at E. So both men were correct.  This was the little joke of the puzzle, which is not by any means difficult.

249.—­THE SAILOR’S PUZZLE.

[Illustration]

There are only four different routes (or eight, if we count the reverse ways) by which the sailor can start at the island marked A, visit all the islands once, and once only, and return again to A. Here they are:—­

A I P T L O E H R Q D C F U G N S K M B A A I P T S N G L O E U F C D K M B Q R H A A B M K S N G L T P I O E U F C D Q R H A A I P T L O E U G N S K M B Q D C F R H A

Now, if the sailor takes the first route he will make C his 12th island (counting A as 1); by the second route he will make C his 13th island; by the third route, his 16th island; and by the fourth route, his 17th island.  If he goes the reverse way, C will be respectively his 10th, 9th, 6th, and 5th island.  As these are the only possible routes, it is evident that if the sailor puts off his visit to C as long as possible, he must take the last route reading from left to right.  This route I show by the dark lines in the diagram, and it is the correct answer to the puzzle.

The map may be greatly simplified by the “buttons and string” method, explained in the solution to No. 341, “The Four Frogs.”

250.—­THE GRAND TOUR.

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.]