Solution

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There are several persons mentioned in the problem whose relationship is essential to its solution. Let call the fellow in the portrait A and the one looking at A's portrait B. The B's sentence, the gist of the problem, can be translated as

There is no doubt that, since B has no siblings, he is the only child of his father. In other words, B's father's son is . (1) then admits of simplification:

which is another way of saying that A is B's .

This was simple, but what if B said instead, "This man's son is my father's son." How would the two be related? Let's translate:

which, as we already know, reduces to

This exactly means that A is B's . And what would we learn from a slightly different sentence, "This man's son is my son's father"? This one is still translated as

because "my son's father" is . So this problem has exactly same solution as the previous one. And coming full circle, what do we get from "This man's father is my son's father"? It is rather obvious that "this man" and "my son" have the same father, so that A is B's son as in the original problem.

References