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One day, while enjoying a trip in one of his kingdom's cities, the amnesiac king of Country X forgets the identity of his sole son and heir. Three men from the empire rush to send letters to the king--all claim to be of royal blood.

First claimant:

Father; do you remember the days we spent playing chess against each other? We spent entire afternoons with only each others' company honing each others' skills.

Second claimant:

Your majesty; do you remember the speech we gave together at the city hall? You praised the younger me for having the courage to speak in public.

Third claimant:

My liege; do you recall that day on late Mother's birthday, when I broke her favorite antique vase? To teach me a lesson, you made me clean the mess myself.

After inspecting all three letters, the king visits the royal library to read, as he often does. When he returns to his chambers that night, he quickly pens a letter to one of the claimants and has the other two arrested. Which person does he decide is the prince?

I believe that it is...

the second claimant.

Why?

You say that the king went to the royal library, where he found out who the correct claimant was. The first and third claimants both refer to personal incidents, which probably weren't published in any book. However, the speech that the king and his son gave was probably published in some book that the royal librarian has stocked, so after reading that book and seeing the speech, he remembers that he and his son gave that exact speech printed there.

I feel like this riddle may be a tad too vague but I'm going with

The first claimant.

Why:

Because he went to the library and saw he had dozens and dozens of chess books in his private collection. Something only those close to him would know. The speech is a public incident that many people knew about so any of them could claim it for themselves

I think he decides that the

third

claimant is telling the truth. My reasoning:

The king has amnesia, but the first and second claimants refer to scenarios where the king must have used his memory: Improving at chess and rehearsing a public speech.

I feel like it is

The second claimant

Because

He uses "Your Majesty" and refers to himself as "the younger me", both with the air of formality that one might expect from royalty.

Also

The Third claimant uses "My Liege" which is a feudal relationship between lord and vassal. He also refers to "Mother" with no royal title

And

The First claimant uses "Father", which is rather informal for a royal to use in a letter.

I think the heir is the

third claimant. The king is amnesiac so he wouldn't have gotten better at chess through practice.
He wouldn't remember speeches either for the same reason. But the third claimant claims he broke a vase. That vase could've been placed in the library where the king spends large amounts of time. The king would've have remembered the vase, and then it would be easy to confirm with any of his servants whether the vase-breaking incident had happened or not.