Posted elsewhere is the problem of the jumping frog. Geogebra makes the problem easy and shows that it is more geometric than anything else.
A bullfrog leaps 2 meters in some direction. Does not like its location so it randomly leaps n meters again in some direction. If its odds of being within 1 meter from its original position are 1 / 6 then what is n?
Basically what you do is draw a unit circle around the origin. The origin represents the starting point of the frog. Draw another concentric circle with radius 2 (shaded circle). This represents all possible first jumps. WLOG pick a point on the shaded circle and call it B. Draw two tangents from B to the smaller circle. Call the tangent points F and G. Now it is just a geometry proof. Angle FBG is 60 degrees. The red circle represents all possible 2nd jumps. This angle of 60 degrees represents 1 / 6 of all possible second jumps.
Line segments AB and BC arelong. That is the length of n.
In mathematics, you don't understand things. You just get used to them.
I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.