How can the night be divided to sleep half of it, stand (in prayer) for a third of it, and sleep a sixth of it, in emulation of David (peace be upon him), when the times of the 'Isha and Fajr prayers are known?

The calculation of the night begins from sunset until the break of dawn. Midnight is approximately at 11:55, and the third third of the night begins after the end of the first half until approximately 3:45, which is the time for night vigil mentioned in the hadith about the night vigil of the Prophet Dawud, peace be upon him. From the time it ends until the break of dawn is the sixth sixth of the night, which is the time to return to sleep.