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!


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.


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 12 of each other?


The following picture is the set of points (x,y) such that |xy|12:


So the probability is 34.

Source : Link , Question Author : Jeel Shah , Answer Author : Peter Woolfitt

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