Question

Did Pandavas really complete their ajnatavasa?

Answer

At first, as per Duryodhana, they didn't complete:

Vaisampayana said, 'King Duryodhana then, on the field of battle said unto Bhishma, and unto Drona--that tiger among warriors, and unto Kripa--that mighty car-warrior, these words, 'Both myself and Kama had said this unto the preceptors 1 I refer to the subject again, for I am not satisfied with having said it once. Even this was the pledge of the sons of Pandu that if defeated (at dice) they would reside to our knowledge in countries and woods for twelve years, and one more year unknown to us. That thirteenth year, instead of being over, is yet running. Vibhatsu, therefore, who is still to live undiscovered hath appeared before us. And if Vibhatsu hath come before the term of exile is at end, the Pandavas shall have to pass another twelve years in the woods. ~ Mahabharata: Virata Parva: Go-harana Parva: Section XLVII

But, later Bhishma explained and as per this explanation, they completed:

Bhishma said, 'The wheel of time revolves with its divisions, viz., with Kalas and Kasthas and Muhurtas and days and fortnights and months and constellations and planets and seasons and years. In consequence of their fractional excesses and the deviations of also of the heavenly bodies, there is an increase of two months in every five years. It seems to me that calculating this wise, there would be an excess of five months and twelve nights in thirteen years. Everything, therefore, that the sons of Pandu had promised, hath been exactly fulfilled by them. Knowing this to be certain, Vibhatsu hath made his appearance. ~ Mahabharata: Virata Parva: Go-harana Parva: Section LII

This is explained in Mahabharata - SanskritHindi version from GitaPress by Pandit Ramnarayan commentary in more detail:

Mahabharata - SanskritHindi version from GitaPress by Pandit Ramnarayan - commentary1 Mahabharata - SanskritHindi version from GitaPress by Pandit Ramnarayan - commentary2

The English translation (as per my understanding) of above screenshot would be as below:

  • Lunar year: 354 days + few hrs
  • Solar year: 365 days + few hrs

13 Solar years = (365 * 13) = 4745 days + few days

13 Lunar years = (354 * 13) = 4602 days + few days

So there is a difference of around 5 months in both the calculation. As per Pandit Ramnarayan's commentary, Pandavas considered Solar calculation, so as per them, they just completed 13 years. But, Bhisma considered Lunar calculation along with Adhika masa, so as per him Pandavas exceeded 5 months and 12 days after completing 13 years.

By this explanation, Pandavas completed agreed 13 years as per both Solar and Lunar (including & excluding Adhika masa) calculations.

As mentioned in below answer, one who believe in Lunar calculation mentioned that Pandvas completed 13 years term even before [5 months] Arjuna reveals his identity. But Pandavas themselves were believing Solar calculation so they appeared when they completed 13 years in every calculation.

As you can see, difference is due to Adhik Masa, Lunar calculation and Solar calculation. After, Bhisma's explanation Duryodhana somehow seems to be convinced and instead of sending them again in woods for 12 years, he prepared to fight:

"Duryodhana said, 'I will not, O grandsire, give back the Pandavas their kingdom. Let every preparation, therefore, for battle be made without delay.'

UPDATE

Also, as discussed in Did prince Yudhishthira obtain boons from Yama for successful completion of Agyatvaas? post, Yudhishthira obtained a boon form Yama for 'completion of thirteenth year, secretly and unrecognized':

Vaisampayana continued,-'Thereat that worshipful one replied,--'I give this boon unto thee!' And then reassuring Kunti's son having truth for prowess, he also said, 'Even if, O Bharata, ye range this (entire) earth in your proper forms none in the three worlds shall recognise you. Ye perpetuators of the Kuru race, through my grace, ye will spend this thirteenth year, secretly and unrecognised, in Virata's kingdom!

So, as per the Yama's boon also, they must have completed the 13th year unrecognized.

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