Disappearing Bicyclist – Sam Loyd (1906)

My guess:

The problem is one of discrete values. You are asking about "a boy", but there are no "a boy"s. There are only fractions of boys. The total sum of "boy fractions" is the same, but the number of pairs of "boy fractions" that meet the distinction of "a boy" changes. "A boy" does not vanish because there were never "a boy"s in the first place.

One might have better luck with "boy heads", and you can see that at roughly 5 oclock there is a "boy head" that turns into a "boy arm". So part of the trick is that some of the "boy fractions" change in your interpretation. Without the trick, you would reasonably say, "Wait, there is still a boy head there! So there are still 13 boys!"

in a sense, there are only two boys, an inside boy and an outside boy, and where those two boys are together (at around 8 o'clock) you actually don't see 100% of either of them, yet you count it as two boys

in the A position, look at around 8 oclock where the two boys next to each other.

look at the "inside" boy, and then move clockwise and you just see just his foot on the inside, continue and you see a leg, going all the way around you can see almost all of the "inside boy" emerge.

in a like manner, look at the outside boy of the two boys together, and proceeding counter-clockwise you see the foot, the leg, etc as almost all of the outside boy emerges.

it reminds me of a trick where you take a stack of dollar bills and slice a thin strip from each one, with the slice taken on each successive bill moving across, and tape each bill together again.

At the end, tape all the little strips together and you have an extra bill!

Simple, the "B" boys each average 1/13 bigger.

I also couldn't help but notice this after reading your comment, but in the image it seems pretty clearly that the A configuration is pretty incoherent. The 5 o'clock boy you mentioned, for example, clearly has a sleeve for the right side of his face, the boy at 2 o'clock is also missing a large chunk of his head, and of course the two boys at 8 o'clock overlap, but, now that I look again, in an incoherent way. To me it seems that the answer to where the boy goes is simply that "the A configuration is invalid, but slightly so that we might not notice"

The little line at the top is actually an interactive slider control so you can rotate the inner disk from point A to point B.

Definitely not an intuitive interface.