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Iterated Specs (Posted on 2010-05-28) Difficulty: 2 of 5
There is a number N such that if you inscribe an N-gon in a circle, with all its angles an equal number of integral degrees, that N-gon will necessarily be a regular N-gon, with equal side lengths.

Also, if M is the number of possible divisors of N, including 1 and N itself, then the number M also meets the same criteria: If you inscribe an M-gon in a circle, with all its angles an equal number of integral degrees, that M-gon will necessarily be a regular M-gon, with equal side lengths.

What are N and M?

  Submitted by Charlie    
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Solution: (Hide)
The regular polygons with an integral number of degrees in their angles are as follows:

degrees      sides     divisors of
                     number of sides
60            3             2
90            4             3
108           5             2
120           6             4
135           8             4
140           9             3
144           10            4
150           12            6
156           15            4
160           18            6
162           20            6
165           24            8
168           30            8
170           36            9
171           40            8
172           45            6
174           60            12
175           72            12
176           90            12
177           120           16
178           180           18
179           360           24

In order for a regular polygon to be the only inscribable polygon of that number of sides and equal angles, the number of sides must be odd, as an even number of sides could have two alternating side lengths and still have equal angles. The only entry on the above table with an odd number of sides which also has an odd number of divisors, is the 9-gon.

So N is 9 and M is 3.

DEFDBL A-Z
PRINT
FOR d = 60 TO 179
  den = 180 - d
  IF 360 MOD den = 0 THEN
    n = 360 / den
    ct = 0
    FOR i = 1 TO n
      IF n MOD i = 0 THEN ct = ct + 1
    NEXT
    PRINT d, n, ct
  END IF
NEXT

Based on Enigma No. 1587, "Deja vu", by Susan Denham, New Scientist, 20 March 2010.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
Aha!Steve Herman2010-05-28 16:15:33
Some ThoughtsMore answers? (SPOILER?)Steve Herman2010-05-28 16:06:03
Solutionanother approachDej Mar2010-05-28 13:26:20
number theory approachDaniel2010-05-28 12:26:47
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