Can you solve this 6th grade math problem?

[Alex K]

What quantum divergences are you referring to? 

Are you (a) being serious with this, or (b) is it only a joke?

My bolding.

The quantum answer is b.

The answer is a quantum superposition of (a) and (b): It's a silly joke with a relatively deep background in mathematics and physics.

My bolding.

The answer is b.

That's too bad if that is the case. He gave a particularly intriguing answer, I was looking forward to him explaining it.

It is a joke, but it has enough real content that you shan't be disappointed:

In quantum field theory, many quantities such as the masses of particles and the strength of particle interactions at first come out as infinite, and upon closer inspection one realizes that this "infinity" appear because one has missed a crucial feature: these quantities are not fixed numbers, but depend on the scale at which one is examining the physical system.

For example, the strength of the so-called "strong force" gets weaker, the higher the energy of the collision is. A very good understandable analogy is given by fractals such as the length of a shoreline. The more fine-grained the scale is at which you measure the length of a shore line, the longer it becomes, and if you describe the shoreline by a mathematical fractal, the length actually goes to infinity if you measure it down to infinitely small scales. The properties of particles in quantum theory, it turns out, behave very similarly.

So my joke (I love explaining jokes) consisted of taking something as pedestrian as the geometric shapes in the OP and acting as if we were dealing with subtle quantum phenomena. However, this joke might not be all that far from the truth if you really look at it: If you think of the shapes in the OP as something actually drawn on a piece of paper, there are two effects which go in the direction of my facetious comment.

Once you examine an actual drawing down to small scales, the coarse molecular nature of the paper and the ink will become apparent and the true definition of what the area is becomes fuzzy. On a more fundamental level, if you think of the shapes in the OP as something that is actually sitting in physical space (as opposed to being a mathematical abstraction), once one examines the quantum nature of space itself, the same weird scale-dependence of measured values (such as areas) will likely show up again, just like they appear with particles in quantum theory, or with the length of fractal shorelines. The problem is namely that once you try to examine the area of the drawing down to the Planck scale, the quantum fluctuations of space itself can let the area appear infinitely large if you look close enough, because the surface of the drawing might become fuzzy, a bit like a fractal, due to the quantum uncertainty in the geometry of space itself.