So, you're hiking along the trails on the bluffs of La Crosse letting your mind wander. Certain patterns in nature come swirling through your thoughts. You pause and see two sticks next to each other on the forest floor and remarkably they're almost twins. Most important, they're exactly the same length. Your wandering mind asks, where would I have to break one of those sticks so that the length of intact stick divided by the length of the big piece of the broken stick is the same as the big piece divided by the small piece? And what would be the significance of the answer to this question? You sit down on a nearby rock and think through the problem. Then, in a golden moment you have it.

What is your answer to to this puzzle? Where would you break your stick? In half? No, because the intact stick divided by half a stick would be two and the half stick divided by the other half would be one. Not equal.

Hint- the square root of 5 is part of the solution.

Feel free to post answers in the comments.

The golden ratio is 1.618, I think, and objects that use the golden ratio are considered to be more esthetically pleasing than others, right? The golden ratio is found in nature in some seedheads, shells, and elsewhere: https://memolition.com/2014/07/17/examples-of-the-golden-ratio-you-can-find-in-nature/. Related to the Fibonacci sequence.

If I consider the length of both original sticks to be 1, then after the second stick is broken the length of the two new sticks will be x and 1-x. Since the ratio of stick lengths are equal, then 1/x = x/(1-x), or 1= (x*x)/(1-x), or 1-x=(x*x), or (x*x)+x-1=0. Now to dig up the Quadratic Formula, x=(-b+sqrt(b*b-4ac))/2a. In this case, a=1, b=1, & c=-1. So x=(-1+sqrt(1+4))/2 = (-1+(sqrt5))/2 = (-1+2.236)/2 = 1.236/2 = .618

So what's the significance of .618....?