# An example of a problem which is difficult but is made easier when a diagram is drawn

I am writing a blog post related to problem solving and one of the main techniques used in problem solving is drawing a diagram. Essentially, I want to illustrate that some hard problems (for example, word problems) can be done fairly easily when diagrams are drawn.

There is a paper related to this question but the problems in that paper are much to simple. I was considering using the problem statement of the Langley’s Problem but that question is not too easy even after the diagram.

Does anyone have any ideas?

Thanks a lot!

EDIT:

As pointed out in the comments, it’s difficult to define an “easy” problem or a “difficult” problem. So, I would like to add that the best examples would be within the undergraduate mathematics range or word problems that non-math majors/mathematicians would understand.

EDIT 2:

I would like to clarify further (thanks to @ruakh) that “when a person trying to solve the problem draws a diagram” is what I am looking for as opposed to “when a person who already knows the solution draws a diagram that illustrates it”

Question: Choose two points uniformly at random on the unit interval. What is the probability that these points are within distance $\frac{1}{2}$ of each other?
The following picture is the set of points $(x,y)$ such that $|x−y|\leq\dfrac{1}{2}$:
So the probability is $\dfrac{3}{4}$.