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## The Frogs And Tumblers

(THE PROFESSOR'S PUZZLES)

"What do you think of these?"

The Professor brought from his capacious pockets a number of frogs, snails, lizards, and other creatures of Japanese 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 illustration, he placed eight of the little green frogs on the glasses 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 glasses, 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," anticipated the Professor. "No; the frogs do not exchange positions, but each of the three jumps to a glass 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."

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