259.—­ HANNAH’S PUZZLE.

Starting from any one of the N’s, there are 17 different readings of NAH, or 68 (4 times 17) for the 4 N’s.  Therefore there are also 68 ways of spelling HAN.  If we were allowed to use the same N twice in a spelling, the answer would be 68 times 68, or 4,624 ways.  But the conditions were, “always passing from one letter to another.”  Therefore, for every one of the 17 ways of spelling HAN with a particular N, there would be 51 ways (3 times 17) of completing the NAH, or 867 (17 times 51) ways for the complete word.  Hence, as there are four N’s to use in HAN, the correct solution of the puzzle is 3,468 (4 times 867) different ways.

260.—­THE HONEYCOMB PUZZLE.

The required proverb is, “There is many a slip ’twixt the cup and the lip.”  Start at the T on the outside at the bottom right-hand corner, pass to the H above it, and the rest is easy.

261.—­ THE MONK AND THE BRIDGES.

[Illustration]

The problem of the Bridges may be reduced to the simple diagram shown in illustration.  The point M represents the Monk, the point I the Island, and the point Y the Monastery.  Now the only direct ways from M to I are by the bridges a and b; the only direct ways from I to Y are by the bridges c and d; and there is a direct way from M to Y by the bridge e.  Now, what we have to do is to count all the routes that will lead from M to Y, passing over all the bridges, a, b, c, d, and e once and once only.  With the simple diagram under the eye it is quite easy, without any elaborate rule, to count these routes methodically.  Thus, starting from a, b, we find there are only two ways of completing the route; with a, c, there are only two routes; with a, d, only two routes; and so on.  It will be found that there are sixteen such routes in all, as in the following list:—­

a b e c d b c d a e a b e d c b c e a d a c d b e b d c a e a c e b d b d e a c a d e b c e c a b d a d c b e e c b a d b a e c d e d a b c b a e d c e d b a c

If the reader will transfer the letters indicating the bridges from the diagram to the corresponding bridges in the original illustration, everything will be quite obvious.

262.—­THOSE FIFTEEN SHEEP.

If we read the exact words of the writer in the cyclopaedia, we find that we are not told that the pens were all necessarily empty!  In fact, if the reader will refer back to the illustration, he will see that one sheep is already in one of the pens.  It was just at this point that the wily farmer said to me, “Now I’m going to start placing the fifteen sheep.”  He thereupon proceeded to drive three from his flock into the already occupied pen, and then placed four sheep in each of the other three pens.  “There,” says he, “you have seen me place fifteen sheep in four pens so that there shall be the same number of sheep in every pen.”  I was, of course, forced to admit that he was perfectly correct, according to the exact wording of the question.