[math-fun] Kindergarten example of almost-integer mystery
A crossword puzzle with a 23x23 grid is 529/441 times as large as one with a 21x21 grid. 529/441 is almost 1.2, and the difference D satisfies 1/D = 2204.999999999853571+. Its difference from an integer satisfies |2205 - 1/D| < 1.5 e-10 --Dan Those who sleep faster get more rest.
sqrt(30)/5 has 1 + 1/(10 + 1/(2 + 1/(10 + ...))) as its continued fraction expansion. Cutting it off after that 2 gives you 23/21 as the approximation to the square root, and since the next term is pretty big (10) you get a good rational approximation for such a small denominator. I suppose this gives a sort of recipe for finding more "coincidences" like this one. --Joshua On Sat, Feb 5, 2011 at 7:45 AM, Dan Asimov <dasimov@earthlink.net> wrote:
A crossword puzzle with a 23x23 grid is 529/441 times as large as one with a 21x21 grid.
529/441 is almost 1.2, and the difference D satisfies
1/D = 2204.999999999853571+.
Its difference from an integer satisfies
|2205 - 1/D| < 1.5 e-10
--Dan
Wait! Isn't 6/5 - 529/441 = 1/2205 exactly? --ms On 2/5/2011 10:45 AM, Dan Asimov wrote:
A crossword puzzle with a 23x23 grid is 529/441 times as large as one with a 21x21 grid.
529/441 is almost 1.2, and the difference D satisfies
1/D = 2204.999999999853571+.
Its difference from an integer satisfies
|2205 - 1/D|< 1.5 e-10
--Dan
Those who sleep faster get more rest.
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participants (3)
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Dan Asimov -
Joshua Zucker -
Mike Speciner