Why does the Buddha always answer a question if asked three times?
It's fairly clear that three times was a social convention for crossing some sort of line; either that or just a convention of the Buddha. See also DN 3:

‘Reverend Gotama, there are four castes: the Khattiyas, the Brahmins, the merchants and the artisans. And of these four castes three — the Khattiyas, the merchants and the artisans — are entirely subservient to the Brahmins. With regard to this, it is not proper that they should not pay homage to the Brahmins.’ This was the third time Ambattha accused the Sakyans of being menials.
Then the Lord thought: ‘This young man goes too far in abusing the Sakyans.&quot;

and

Again Ambattha remained silent, and the Lord said: ‘Answer me now, Ambattha, this is not a time for silence. Whoever, Ambattha, does not answer a fundamental question put to him by a Tathāgata by the third asking has his head split into seven pieces.’

Walshe, the translator of the above, gives as a note for the second passage, &quot;A curious threat that (as RD observes) never comes to anything, and is of course pre-Buddhist.&quot;
Not sure if the &quot;third time&quot; is also pre-Buddhist, but again, it appears to be some sort of line in the sand for the Buddha at least.