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Those Puzzling Trees

Updated: Apr 29, 2022

OK all you math whizzes out there, here's your second puzzle. Remember the first one? It was in a previous blog post where two sticks magically led you to the value of the Golden Ratio.

Pretend you're nursing along a bunch of small trees that were planted a few years ago on land that used to be an old corn field. Like the land called Skemp 2 on the corner of F and FA on the way to Upper Hixon. The trees are mostly oaks and are planted in rows. I know, they look kind of goofy in rows, but it's hard to plant them otherwise, and that's what happened before you took over. Your trees are pretty close together, and you know you'll eventually have to thin them out to roughly 40 feet apart so they will be able to expand their crowns to about 40 feet before touching the branches of the next tree. Here's a picture of circles meant to represent the small trees from above before the crowns grow to 40 feet in diameter:

This next picture is when they have achieved that full 40 foot crown and are just starting to bump into each other:

I know, trees don't grow in perfect circles, but let's pretend that they do for this puzzle. And who knows, maybe they do have some strange powers of communication, as in Peter Wohlleben’s book The Hidden Life of Trees, What They Feel and How They Communicate, and somehow “respect“ each other’s territory. As you can see, if they did grow in circles and stopped once they touched, there would be an area between the trees that would be left open as in the green part of this picture:

Next, pretend that a smaller tree that got a late start grew into this area and mostly filled in the gap before butting up against the larger trees like in this picture:

Now, here's your math quiz- The "R" in the last picture is the radius of the crowns of the large trees and equals 20 for a total 40 foot crown. Your first task is to find the radius of the little tree that filled in the gap, the "r" in the diagram. If R=20 feet, what is r? The second task is to find how much space is left for sunlight to still get through, as seen in green space in the last picture after the little tree grows. What is the area in square feet of the green in the last picture?

Don't despair! You can do it! Look at the pictures carefully, remember some of your high school math, and find that little r. Then, for a bonus, find the area left over in green.

Please leave your answers in the comments section and explain how you derived them. The first correct answer that is posted will get a bottle of Rigden maple syrup produced in the spring of 2021 the old fashion way (board members not eligible). Very yummy!

If no correct answer comes in after... well, after a while, maybe a month or so, I'll explain how I came up with the answers in a subsequent post and that lonely bottle of maple syrup have to stay home. Good luck!

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Jim Rogala
Jim Rogala
Nov 11, 2021

Here's my stab at it (your initial diagram with the square made it easier):

  1. The small circle in the middle would have a radius of 8.284, derived from the diagonal of a 20x20 square (28.284) minus the R=20.

  2. The area unaccounted for by the circles (space not in canopy) is 127.77, derived from the area of the whole square (1600) minus the area of the large trees (1256.64 from 4 1/4 trees of R=20) minus the area of the small tree (215.54 from r=8.284).

Probably not the most elegant, but I think it is right.


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