Professor Stewart's Hoard of Mathematical Treasures Hardcover – Oct 8 2009
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For the high school student, much of the material in the book will probably be hard going, but the great thing about the book is that it is so full of fascinating problems and diversions that it is necessary only to turn a page or two to get to something more congenial to the reader. For the more advanced college math major, there is much here to educate and delight.
To give but a single example of the mathematical puzzles the book deals with, I will refer to Professor Stewart's treatment of the sequence "1, 11, 21, 1211, 111221, . . ." In this sequence each term after the first is constructed by "reading" the previous term. Thus, the fourth term reads "one 1, one 2, two 1s" and thus generates the fifth term. At first glance, there seems to be little mathematical about this sequence. It's more of a cute brain teaser that really has little to do with math. But what if we asked how many digits the nth term has? Professor Stewart presents that response and an approximation. (In this case, he does not explain the derivation of the approximation, but the point is that he does go well beyond the standard treatment of the sequence.)
My favorite parts from 'Hoard' -
* Byron on Newton
* Egyptian Fraction / Greedy Algorithm
* Fermat's poser (Euler's solution which was not mentioned; stopped after 3 hrs of furtive trying)
* Short article on ancient Indian mathematician Lilavati
* Proof techniques. e.g., Proof by vague authority -- 'Ortho-equilibrium in Cauchy-Mandelbrot set disavows Deric fallacy' -- made it up, but you get a gist!
* Invention of '=' (equals to) sign
* and..the parlor trick of three inverted cups (could make one mad just trying in frustration)
I am still reading the book (as many topics tease me to do further 'research' myself, I proceed only slowly), but it is really great fun for all who like math puzzles and the like. My kids now do find me a real math geek even more - though I am not (I think...) - but the book surely will get a prominent place on my bookshelves. If I will finish the book, I might buy some other books of Ian Stewart as well.