The theme of 52! Fifty-Two Factorial by Michael Wayne Cottle is his search for a concrete comparison that will impress upon his readers just how unthinkably large the number of arrangements of a fifty-two card deck of ordinary playing cards really is.
No spoiler alert required. Other than to praise his work, I will relate only a minimum of Mr. Cottle's spectacular results, for leaks would bleed off some of the rightful effect the author has ciphered hard to achieve.
Authors often promise right away there is not much math within. Cottle does not do this, and there is quite a lot of math within. Though it is all elementary math, it took practice and thought, and Cottle finds new slants (at least some were to me)—teaching, amazing and entertaining simultaneously.
It is worth noting that he is an author who has published fiction as well as this work of non fiction. Being a professional writer is perhaps why he was aware enough in the first place to avoid many common clichés of the genre.
Mr. Cottle certainly scooped me, for I had nearly the same book in mind and some preliminary calculations in hand myself (whose results I found surprising) before learning someone named Cottle had just published (in 2013) such a work. Fortunately for the world, he beat me to press. I say this because Mr. Cottle has done an excellent job which kept me thoroughly entertained. It took me five years to finally break down and buy our book—I mean his book, but I was glad I did for more reasons than mere relief of curiosity.
The book's oft repeated litmus test is in comparing the magnitude of 52! to things whose size we already have a notion of, or things we can envision the size or scope of. The fact that it even has a central theme immediately sets it apart from many similar books of number curios, and its title is guaranteed to interest card players.
The author's method is to proceed through repeated examples of failure, finding again and again that another physical metric has proven inadequate to generate, accumulate or accommodate 52! of something. He starts with the number of miles in the diameter of the known universe, then reduces that metric to lining up eggs across the same distance, and proceeds to finer scales from there. Along the way he manages to compare his book's central problem to a host of tireless workhorses and paradoxes routinely found in most books of number curios, from chessboard rice doubling to monkeys randomly typing Shakespeare. This ability to be entertaining and to offer new insights while covering well worn ground is an art in itself.
Cottle has a talent (and an obvious passion) for calculating, which is on full display in his 144 page text. He is not bluffing, and understands what he is up to. He says he only has a masters in mathematics. (His only, not mine.) He possesses a thorough understanding of every term in the equations he utilizes, and gives precise explanations of what each term means. At times his fractions are displayed in English instead of numerals for ease of clarity when he wants concepts absorbed. Though all calculations in the book are elementary, this does not mean simple. The author's grasp of elementary physics and its army of units seems quite good as well.
Stylistically, Cottle avoids most of his subject's tired clichés. This is quite refreshing, and an achievement, since his subject is near the heart of amateur mathematicians' preferred country, so has been mined redundantly. My guard always up, at first I thought I detected the beginnings of the overuse of patronizing humor many authors on popular numbers habitually employ to try and set their audience at ease. However, bad humor where rigorous concentration would be more appropriate is not an indulgence of Mr. Cottle's, at least not one he overuses. Had my original suspicion proven correct, I probably would never have finished his little title, which tells you I did finish it.
Once all the calculating is done, Mr. Cottle feels like philosophizing. His conclusions are so similar to my own independently achieved conclusions that it grows eerie. A person likes their own thoughts, and I found plenty in Cottle. I also found plenty I hadn't thought of before but wished I had. I found things I should have known but didn't, and corrected them.
Though 52! Fifty-Two Factorial, is exactly what I wanted to read and reflect on, I would recommend this book to others of numeric or philosophic persuasion. Some of the author's calculational results are gems. I have wondered if any were achieved strictly through symbol manipulation or if all were derived numerically on an apparatus. He knows many amazing things. For instance he knows that
P(70, 41)>52!, but not by much. He knows 40 at a time would be too few.
Or he knows that P(2000, 21)>52!, but just barely. (Permutations of 2000 objects taken 21 at a time is the English translation of this inequality.)
I found this book to be well worth my time spent.
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