By Martin Gardner

The twenty chapters of this booklet are well balanced among every kind of stimulating principles, urged by way of down-to-earth items like fit sticks and greenback debts in addition to by means of far off gadgets like planets and limitless random walks. We know about historic units for mathematics and approximately glossy factors of man-made intelligence. There are feasts the following for the eyes and palms in addition to for the mind.

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Sample text

Letting x be the radius of the orange, we write the equation, which gives x a value of one inch. The problem can, of course, be solved in other ways. When it appeared as problem 43 in the Pi M u Epsilon Journal, Nbvember 1952, Leon Bankoff solved it this way, with R the radius of each large sphere and r the radius of the small sphere: "The small sphere, radius r, touches the table at a point equi- distant from the contacts of each of the large spheres with the table. Hence it lies on the circumcenter of an equilateral triangle, the side of which is 2R.

Black and gray matches can also be used for playing Connecto, a game described by David L. Silverman in his book Your Move (McGraw-Hill, 1971). Here too the players alternate in placing matches on a square matrix of any size, but the object now is to be the first to enclose a region of any shape within a boundary of one's own matches. In Figure 14 Black has won the game. Can you discover Silverman's simple strategy by which the second player can always prevent the first player from winning, even on an infinite matrix?

As the philosopher Karl Popper maintains, the "strongest" conjecture is the one that is easiest to falsify, and Popper considers this the equivalent of the "simplest" conjecture. In Sackson's game the strongest (and simplest) conjecture is that every cell contains the same symbol, say a star. It is strong because a single inquiry about any cell, answered by anything but a star, falsifies it. The weakest conjecture is that each cell contains one of the four symbols. Such a hypothesis can be completely confirmed.