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 Just a fraction off (Posted on 2006-05-28)

Egyptian Fractions are fractions expressed in the form of 1/n or as the sum of such fractions (1/a + 1/b + ...). For example, 2/3 would be expressed as 1/2 + 1/6.
(As found on the Rhind papyrus, 2/3 was one, if not the only, non-unit fraction used by the Egyptians.)

The following Egyptian Fractions correspond to the fractional portion of the first ten non-integer members of a "regular" series. (The "whole number" component of each member of the series is omitted and any series members without fractional components is likewise omitted).

What is the missing fraction in this sequence?

1/2 + 1/14
1/4 + 1/44
1/4 + 1/18 + 1/468
1/4 + 1/28
1/2
1/2 + 1/4 + 1/14 + 1/476
1/19
1/2 + 1/3 + 1/42
1/2 + 1/8 + 1/88
1/3 + ?

Submitted by Dej Mar
Solution: (Hide)

1/69. The ten unit fraction sums are the Egyptian Fractions of the fractional part of the angle in degrees of the first ten Regular Polygons that have a fractional part.

Following is the list of the first 23 polygons ordered by the number of sides, their angles, and their angles represented with Egyptian Fractions:
 3 triangle 60 60 4 square 90 90 5 pentagon 108 180 6 hexagon 120 120 7 heptagon 128 4/7 128 1/2 + 1/14 8 octagon 135 135 9 nonagon 140 140 10 decagon 144 144 11 hendecagon 147 3/11 147 1/4 + 1/44 12 dodecagon 150 150 13 tridecagon 152 4/13 152 1/4 + 1/18 + 1/468 14 tetradecagon 154 2/7 154 1/4 + 1/28 15 pentadecagon 156 156 16 hexadecagon 157 1/2 157 1/2 17 heptadecagon 158 14/17 158 1/2 + 1/4 + 1/14 + 1/476 18 octadecagon 160 160 19 enneadecagon 161 1/19 161 1/19 20 icosagon 162 162 21 icosihenagon 162 6/7 162 1/2 + 1/3 + 1/42 22 icosikaidigon 162 7/11 162 1/2 + 1/8 + 1/88 23 icosikaitrigon 164 8/23 164 1/3 + 1/69

 Subject Author Date answer K Sengupta 2008-02-26 06:01:58 Correction to the problem Gamer 2006-05-31 00:54:50 re: The busy work brianjn 2006-05-30 03:29:24 re(2): The busy work brianjn 2006-05-30 02:44:14 re(2): The busy work Mindrod 2006-05-29 12:49:49 re: The busy work brianjn 2006-05-29 08:20:45 The busy work Jyqm 2006-05-28 13:13:18

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