By David Gale (auth.), David Gale (eds.)
For these enthusiastic about the summary universe of arithmetic, David Gale's columns in The Mathematical Intelligencer were a major resource of leisure. the following Gale's columns are gathered for the 1st time in ebook shape. inspired through the magazine's editor, Sheldon Axler, to jot down on no matter what happy him, Gale ranged all over the place around the box of arithmetic yet often lower back to favourite subject matters: triangles, tilings, the mysterious houses of sequences given via basic recursions, video games and paradoxes, and the actual automaton that provides this assortment its name, the "automatic ant". the extent is appropriate for people with a few familiarity with mathematical principles, yet nice sophistication isn't needed.
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C? 48 TRACKING THE AUTOMATIC ANT Case II. The third point x or z is not between a and p. Then, as before, either bx or bz will intersect R in a point closer to p than q. (See Fig. ) Voila! 3. Applying Pasch's axiom to the other case. BUT There are 2n points in the plane in general position; n of them are red and n are blue. Show that they span n disjoint segments having one end red and the other blue. Or course, by now you have gotten the message and realize that you should choose segments in a way that minimizes the sum of their lengths.
Thus, for example, if you want to know the maximum rather than the sum of the salaries, the best you can achieve is "minority privacy:' which means that there is a protocol such that no set of fewer than half of the participants can learn anything about the complementary set by pooling information. Indeed, there is what might be called the fundamental theorem of protocols, which asserts roughly that any function can be computed minority-privately, but only a few, like the sum, can be computed n-privately.
Chapter 7 Pop Math and Protocols MORE ON PROTOCOLS: PLAYING GAMES OVER THE TELEPHONE For almost all of what follows I am indebted to Imre Barany. 1. Coin-Tossing A mathematician in Moscow and another in New York are talking long distance about a joint paper they have written which they have been invited to present at a meeting in Paris. Because both of them would like to go but only one can, they decide that the only fair thing is to leave it to chance. "All right;' says the Russian, "I just tossed a coin.