He made a rough diagram, and placed a crown and a florin in two of the divisions, as indicated in the ill.u.s.tration.

"Now," he continued, "place the fewest possible current English coins in the seven empty divisions, so that each of the three columns, three rows, and two diagonals shall add up fifteen shillings. Of course, no division may be without at least one coin, and no two divisions may contain the same value."

"But how can the coins affect the question?" asked Grigsby.

"That you will find out when you approach the solution."

"I shall do it with numbers first," said Hawkhurst, "and then subst.i.tute coins."

Five minutes later, however, he exclaimed, "Hang it all! I can't help getting the 2 in a corner. May the florin be moved from its present position?"

"Certainly not."

"Then I give it up."

But Grigsby and I decided that we would work at it another time, so the Professor showed Hawkhurst the solution privately, and then went on with his chat.

68.--_The Postage Stamps Puzzles._

"Now, instead of coins we'll subst.i.tute postage-stamps. Take ten current English stamps, nine of them being all of different values, and the tenth a duplicate. Stick two of them in one division and one in each of the others, so that the square shall this time add up ninepence in the eight directions as before."

"Here you are!" cried Grigsby, after he had been scribbling for a few minutes on the back of an envelope.

The Professor smiled indulgently.

"Are you sure that there is a current English postage-stamp of the value of threepence-halfpenny?"

"For the life of me, I don't know. Isn't there?"

"That's just like the Professor," put in Hawkhurst. "There never was such a 'tricky' man. You never know when you have got to the bottom of his puzzles. Just when you make sure you have found a solution, he trips you up over some little point you never thought of."

"When you have done that," said the Professor, "here is a much better one for you. Stick English postage stamps so that every three divisions in a line shall add up alike, using as many stamps as you choose, so long as they are all of different values. It is a hard nut."

[Ill.u.s.tration]

69.--_The Frogs and Tumblers._

"What do you think of these?"

The Professor brought from his capacious pockets a number of frogs, snails, lizards, and other creatures of j.a.panese manufacture--very grotesque in form and brilliant in colour. While we were looking at them he asked the waiter to place sixty-four tumblers on the club table. When these had been brought and arranged in the form of a square, as shown in the ill.u.s.tration, he placed eight of the little green frogs on the gla.s.ses as shown.

"Now," he said, "you see these tumblers form eight horizontal and eight vertical lines, and if you look at them diagonally (both ways) there are twenty-six other lines. If you run your eye along all these forty-two lines, you will find no two frogs are anywhere in a line.

"The puzzle is this. Three of the frogs are supposed to jump from their present position to three vacant gla.s.ses, so that in their new relative positions still no two frogs shall be in a line. What are the jumps made?"

"I suppose----" began Hawkhurst.

"I know what you are going to ask," antic.i.p.ated the Professor. "No; the frogs do not exchange positions, but each of the three jumps to a gla.s.s that was not previously occupied."

"But surely there must be scores of solutions?" I said.

"I shall be very glad if you can find them," replied the Professor with a dry smile. "I only know of one--or rather two, counting a reversal, which occurs in consequence of the position being symmetrical."

70.--_Romeo and Juliet._

For some time we tried to make these little reptiles perform the feat allotted to them, and failed. The Professor, however, would not give away his solution, but said he would instead introduce to us a little thing that is childishly simple when you have once seen it, but cannot be mastered by everybody at the very first attempt.

"Waiter!" he called again. "Just take away these gla.s.ses, please, and bring the chessboards."

"I hope to goodness," exclaimed Grigsby, "you are not going to show us some of those awful chess problems of yours. 'White to mate Black in 427 moves without moving his pieces.' 'The bishop rooks the king, and p.a.w.ns his Giuoco Piano in half a jiff.'"

"No, it is not chess. You see these two snails. They are Romeo and Juliet. Juliet is on her balcony, waiting the arrival of her love; but Romeo has been dining, and forgets, for the life of him, the number of her house. The squares represent sixty-four houses, and the amorous swain visits every house once and only once before reaching his beloved. Now, make him do this with the fewest possible turnings. The snail can move up, down, and across the board and through the diagonals. Mark his track with this piece of chalk."

[Ill.u.s.tration]

"Seems easy enough," said Grigsby, running the chalk along the squares.

"Look! that does it."

"Yes," said the Professor: "Romeo has got there, it is true, and visited every square once, and only once; but you have made him turn nineteen times, and that is not doing the trick in the fewest turns possible."

Hawkhurst, curiously enough, hit on the solution at once, and the Professor remarked that this was just one of those puzzles that a person might solve at a glance or not master in six months.

71.--_Romeo's Second Journey._

"It was a sheer stroke of luck on your part, Hawkhurst," he added. "Here is a much easier puzzle, because it is capable of more systematic a.n.a.lysis; yet it may just happen that you will not do it in an hour. Put Romeo on a white square and make him crawl into every other white square once with the fewest possible turnings. This time a white square may be visited twice, but the snail must never pa.s.s a second time through the same corner of a square nor ever enter the black squares."

"May he leave the board for refreshments?" asked Grigsby.

"No; he is not allowed out until he has performed his feat."

72.--_The Frogs who would a-wooing go._

While we were vainly attempting to solve this puzzle, the Professor arranged on the table ten of the frogs in two rows, as they will be found in the ill.u.s.tration.