The 9! Mathematical Recreations book series by Francis Gurtowski
9! Mathematical Recreations is a multivolume, mathematical version of the Official Scrabble Dictionary.
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Weight = mass x gravity; if facts had mass then here on Planet Earth this tenfold book-series might weigh some ten tons or more, rather than just 36 pounds, 6 ounces.
A whopping total of 3,761,064 facts are revealed in 9! Mathematical Recreations; for starters, the 362,880 permutations of the Arabic numerals 1, 2, 3, 4, 5, 6, 7, 8, and 9 are enumerated and serve as contextual markers.
9! Mathematical Recreations includes 1,451,520 (4 x 362,880) contextual instances of the single-digit prime numbers 2, 3, 5, and 7. Scanning left-to-right, 9! Mathematical Recreations also includes this inventory of multi-digit specimens:
And I will give you this additional "fun fact" for free: none of the 9! permutations of 1-9 are prime numbers. |
Try this Nine Factorial "party trick" to (a) engage the more cerebral guests and the posers at your party and (b) amuse the onlookers to the mathematical duel.
9! can be played on the back of a table napkin, but it immensely helps to keep 9! Mathematical Recreations handy to settle disputes given how difficult it is to factor even numbers that are really not that long. Go around the room and ask anyone to recite an Arabic numeral (1-9) that has not been cited before. Let us say that you end up with 856342179 which turns out to be a doozy of a permutation of 123456789. Now ask the players in turn to scan 856342179 left-to-right and either (a) retire or (b) recite an alleged prime number in 856342179 that has not been cited before. Every recitation can be challenged by the other players. A recitation that is successfully challenged, on the basis that the recitation is (a) actually a composite number or (b) not a proper substring of the permutation in play, is grounds for the immediate dismissal of the reciter. A bogus challenge is as onerous as a bogus recitation; in which case it is instead the errant challenger who is forthwith dismissed. The last player standing wins the 9! game. |
"Ho hum" you might say.
How many distinct prime numbers can there possibly be in a nine-place permutation? For starters, at least these four: 2, 3, 5, and 7. 123456789 is an underachiever inasmuch as scanned left-to-right 123456789 only has these nine prime substrings: 2, 3, 5, 7, 23, 67, 89, 4567, and 234456789. 856342179 takes the cake with these 20 prime substrings: 2, 3, 5, 7, 17, 79, 563, 421, 179, 8563, 4217, 2179, 63421, 34217, 42179, 342179, 8563421, 5634217, 6342179, and 56342179. Moreover, 856342179 is one of only 1085 permutations of 1-9 that has at least one prime substring of lengths 1-8 digits. 425861379 and 654829137 are tied for second with 19 prime substrings, at least one of each length 1-8. Let me look them up for you in the 9! Mathematical Recreations book series. 425861379: 2, 3, 5, 7, 61, 13, 37, 79, 613, 137, 379, 5861, 58613, 86137, 61379, 425861, 258613, 2586137, and 25861379. 654829137: 2, 3, 5, 7, 29, 13, 37, 829, 137, 8291, 9137, 54829, 82913, 29137, 548291, 6548291, 5482913, 4829137, and 65482913. |
