Each interior angle of a regular R-gon is precisely 59/58 times that of each angle of a regular S-gon, where R ≥ S ≥ 3.
Find the largest possible value of S.
10 for N1=3 to 10000
20 A1=180*(N1-2)//N1
30 N2=N1
40 loop
50 inc N2
60 A2=180*(N2-2)//N2
70 if A2//A1>59//58 then goto 100
80 if A2//A1=59//58 then print N1;A1,N2;A2
90 endloop
100 next N1
finds
S R
sides angles sides angles
59 10260//59 116 5130//29
60 174 120 177
89 15660//89 356 15930//89
110 1944//11 1595 57348//319
114 3360//19 3306 99120//551
116 5130//29 6728 151335//841
117 2300//13 13572 67850//377
Angles are in degrees.
The program remains stuck on 118.
At 118 sides, the angle at each is 180*116/118 = 10440/59. Multiplying this by 59/58 gives 180, an angle impossible in a polygon: it would be an infinitely sided polygon, aka a circle.
117 is the largest possible value of S.
|
|
Posted by Charlie
on 2015-01-15 10:14:13 |
computer assisted solution
