Consider a convex polygon such that:
- Each of its sides correspond to the square root of a positive integer.
- The polygon can be inscribed in an unit circle.
Determine the total number of polygons that simultaneously satisfy the 2 properties mentioned above.
Note: Polygons that are rotations and reflections of each other are considered the same.
The smallest possible side is Sqrt(1) = 1, and the largest possible is Sqrt(4) = 4.
Sqrt(1) subtends 60 degrees
Sqrt(2) subtends 90 degrees
Sqrt(3) subtends 120 degrees
Sqrt(4) subtends 180 degrees
The possible polygons are then:
1-1-1-1-1-1 Hexagon
sqrt(2)-sqrt(2)-sqrt(2)-sqrt(2) Square
Sqrt(3)-sqrt(3)-sqrt(3) Triangle
2-1-1-1
2-sqrt(2)-sqrt(2)
2-1-sqrt(3)
sqrt(3)-sqrt(3)-1-1
sqrt(3)-1-sqrt(3)-1
sqrt(3)-sqrt(2)-sqrt(2)-1
sqrt(3)-sqrt(2)-1-sqrt(2)
sqrt(2)-sqrt(2)-1-1-1
sqrt(2)-1-sqrt(2)-1-1
Hope I didn't miss any
---------------------------------------
Oh well, I see from Charlie's final answer that I did in fact miss one:
sqrt(3)-1-1-1
for a total of 13
Edited on April 29, 2023, 9:56 am
Solution (spoiler)
