Transcription:
How To Confound A Centaur
Centaur: Hold it right there, you can't just ride through my fields uncontested! I'll lose my credibility if we don't battle or something.
Me: That's fair. How about we–
Centaur: And it can't be a pun battle, I heard what you did to the sphinx.
Me: Darn. Okay, what if I beat you in a horse race?
Centaur: Ha! Alright, your funeral. Where's the finish line?
Me: That tree over there. Where's your horse?
Centaur: (gestures at horsey backside) Um...?
Me: You're not a horse, this is a HORSE race. You have to race with an actual horse.
Centaur: You want me... a centaur... to RIDE a regular horse.
Me: That is, linguistically-speaking, what you agreed to.
Centaur: ...
Me: ...
Centaur: I hate you.
Me: That's fair.
----
In case you're serious with the sphinx question: in ancient Greek myth the entrance to the city of Thebes was guarded by a sphinx who would only let you through if you could solve a riddle, otherwise the sphinx would eat you. Which riddle that would be changes from story to story but the most famous one is along the lines of "What creature walks on four legs in the morning, two at noon and three in the evening?". This was eventually solved by king Oedipus who realized it was a human who crawls as a baby, walks on two legs as an adult and needs a walking stick as a third leg when they're old. Depending on which version you read, the sphinx was either so shocked by him solving the riddle that it threw itself from a cliff or was simply slain by him.
omg this part I did not know. I just thought it...let him pass.
But anyway, what does that have to do with the "pun battle" as posed in this story?
Well, you could consider the solution to the riddle a pun though that's quite a stretch. Though there is at least one modern rendition (in German) that directly uses a more pun-ny solution. It does't quite translate to English but I'll try. Basically, Oedipus thinks and thinks until he starts to complain "Ach Mensch..." which is roughly equal to English "Oh boy..." but "Mensch" is literally the German for "human" so it's the right solution and the sphinx has to accept it. If you understand German, I highly recommend checking out this version. It's "König Ödipus" by Bodo Wartke. He plays all roles himself on a mostly empty stage with only a couple of props and it's absolutely hilarious. The sphinx is a lion hand puppet.
Edit: for anyone interested, here's a youtube clip of the scene: https://www.youtube.com/watch?v=DogC57ZJuY8 (German with German subtitles)
The pun works equally well in English. "Aw, man!" has the same meaning as Ach mensch.
Oh right, that would work.