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Just a fraction off (Posted on 2006-05-28) Difficulty: 4 of 5

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    
Rating: 3.3333 (3 votes)
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 triangle6060
4 square9090
5 pentagon108180
6 hexagon120120
7 heptagon128 4/7128 1/2 + 1/14
8 octagon135135
9 nonagon140140
10 decagon144144
11 hendecagon147 3/11147 1/4 + 1/44
12 dodecagon150150
13 tridecagon152 4/13152 1/4 + 1/18 + 1/468
14 tetradecagon154 2/7154 1/4 + 1/28
15 pentadecagon156156
16 hexadecagon157 1/2157 1/2
17 heptadecagon158 14/17158 1/2 + 1/4 + 1/14 + 1/476
18 octadecagon160160
19 enneadecagon161 1/19161 1/19
20 icosagon162162
21 icosihenagon162 6/7162 1/2 + 1/3 + 1/42
22 icosikaidigon162 7/11162 1/2 + 1/8 + 1/88
23 icosikaitrigon164 8/23164 1/3 + 1/69

Comments: ( You must be logged in to post comments.)
  Subject Author Date
answerK Sengupta2008-02-26 06:01:58
Hints/TipsCorrection to the problemGamer2006-05-31 00:54:50
re: The busy workbrianjn2006-05-30 03:29:24
re(2): The busy workbrianjn2006-05-30 02:44:14
Some Thoughtsre(2): The busy workMindrod2006-05-29 12:49:49
re: The busy workbrianjn2006-05-29 08:20:45
The busy workJyqm2006-05-28 13:13:18
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