Outline Mathematics
Logic Problems

Sons and Fathers

Here's a problem to tackle:

Who is this man?

A man was looking at a portrait. Someone asked him, "Whose picture are you looking at?" He replied: "Brothers and sisters have I none, but this man's father is my father's son."

Whose picture was the man looking at?

Solution


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Copyright © 1996-2018 Alexander Bogomolny

Solution

Who is this man?

A man was looking at a portrait. Someone asked him, "Whose picture are you looking at?" He replied: "Brothers and sisters have I none, but this man's father is my father's son."

Whose picture was the man looking at?

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 A,A,B's father is B,A,B's father's son,son,father,sibling.

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 B,A,B. The above then admits of simplification: A,A,B's father is B,A,B, which is another way of saying that A is B's child,father,sibling,child.

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: A,B,A's son is B,A,B's father's son, which, as we already know, reduces to A,B,A's son is B,A,B.

This exactly means that A is B's father,sibling,child,father. 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 A,A,B's son is B,A,B, because "my son's father" is myself,father,myself,sibling,child. 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

  1. R. Smullyan, What Is The Name Of This Book?, Simon & Schuster, 1978

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Copyright © 1996-2018 Alexander Bogomolny

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