Encompassing puzzle
Since some people thought the puzzle I posted yesterday was dumb, I've decided to post a more "intelligenter" puzzle today.
Start with a circle of 1 unit (1 meter, 1 kilometer, 1 AU, whatever). Circumscribe the circle with an equilateral triangle whose sides are tangent to the circle. Now circumscribe a circle around the triangle so that the corners are just touching the circle. Circumscribe this larger circle with a square, with sides again tangent to the circle. Repeat this pattern with a pentagon, then a hexagon, then a heptagon, an octagon...
In other words, every time you draw a circle, encompass it with an equilateral polygon of N+1 sides, where N was the number of sides of the last polygon you drew. It should start looking like this (not quite to scale).
What happens to the radius of the outermost circle as you repeat this an infinite number of times? Note that as we go out, the difference between radii becomes successively smaller.
Hint: Although not all monotonically decreasing series are convergent, this one is. In other words, as you draw an infinite number of circles, the radius is going to approach some maximum value. What is it? Oh, and you might want a calculator for this one.
Solution: Only one winner this time -- Jim. This one is a little tricky to get started. As with most geometry problems, a good picture goes a long way. First, take a look here.
We want to find a relationship between Rn and Rn+1 for an arbitrary point in our series. From the diagram, it looks like we could relate the two sides if we knew the angle a. Fortunately, we do! This is an n-sided polygon, so we can solve for the angle at point c, which is bisected by the radius shown in the figure. Thus, our recursive relationship is Rn+1 = Rn / cos(pi/n). A brute force computation will show this converges to appromximate 8.7 units.
Start with a circle of 1 unit (1 meter, 1 kilometer, 1 AU, whatever). Circumscribe the circle with an equilateral triangle whose sides are tangent to the circle. Now circumscribe a circle around the triangle so that the corners are just touching the circle. Circumscribe this larger circle with a square, with sides again tangent to the circle. Repeat this pattern with a pentagon, then a hexagon, then a heptagon, an octagon...
In other words, every time you draw a circle, encompass it with an equilateral polygon of N+1 sides, where N was the number of sides of the last polygon you drew. It should start looking like this (not quite to scale).
What happens to the radius of the outermost circle as you repeat this an infinite number of times? Note that as we go out, the difference between radii becomes successively smaller.
Hint: Although not all monotonically decreasing series are convergent, this one is. In other words, as you draw an infinite number of circles, the radius is going to approach some maximum value. What is it? Oh, and you might want a calculator for this one.
Solution: Only one winner this time -- Jim. This one is a little tricky to get started. As with most geometry problems, a good picture goes a long way. First, take a look here.
We want to find a relationship between Rn and Rn+1 for an arbitrary point in our series. From the diagram, it looks like we could relate the two sides if we knew the angle a. Fortunately, we do! This is an n-sided polygon, so we can solve for the angle at point c, which is bisected by the radius shown in the figure. Thus, our recursive relationship is Rn+1 = Rn / cos(pi/n). A brute force computation will show this converges to appromximate 8.7 units.