Compute the smallest integer n for which it is possible to draw an n-gon whose vertex angles all measure 167◦
or 174◦.
The lowest n is 32, and it seems there are only five possibilities altogether as shown in the table below.
The n = 32 case is achieved with 24 167-degree angles and 8 174-degree angles. It's assumed that an appropriate set of lengths of sides is possible so the figure will be a closed polygon.
The program seeks ways of achieving a total of 180*n - 360 total degrees using multiples of 167 and/or 174.
for n=5:10000
goal=180*n-360;
have=167*n;
diff=goal-have;
if mod(diff,7)==0
need=diff/7;
if need<=n && need>=0
fprintf('%5d%5d%5d\n',n,n-need,need)
end
end
end
angles
sides 167° 174°
32 24 8
39 18 21
46 12 34
53 6 47
60 0 60
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Posted by Charlie
on 2025-05-22 09:15:22 |
solution
