This is an excerpt from my upcoming book: Calendars of India – Theory and Practice
This video on YouTube gives a summary of this post.
Envious of the wealth of the Pandavas and enraged at the insults from the Pandavas when he visits their palace in Indraprastha, Duryodhana, at the advice of his uncle Shakuni, invites them to a game of dice, in which he defeats the Pandavas (with some dubious help from his uncle). According to the terms of the wager, the Pandavas are forced to go into exile for 12 years and then spend one year incognito. If their identity is discovered within that year, they have to repeat the exile.
During the thirteenth year, the Kauravas make all attempts to unmask the Pandavas. But the Pandavas are living incognito at the court of King Virata and the Kauravas are not able to find them.
Suspecting that the Pandavas may be with King Virata, they force a battle against him, knowing that if the Pandavas were living at Virata’s court, they will come to the aid of Virata in battle and thus would expose the fact that they were indeed the Pandavas.
This is precisely what happens. Arjuna appears as himself and routs the Kauravas. Duryodhana is happy that the Pandavas have been discovered. But Bhishma tells him that the 13-year period has ended and that the Pandavas were safe from further exile.
Duryodhana says (Mahabharata, Critical Edition, Virata Parva, Section 42, Stanzas 1 to 6)
…The thirteenth year is not over. It is still running. But Arjuna who is supposed to remain undiscovered has appeared before us. And since he has appeared before the exile period is over, the Pandavas have to spend another 12 years in exile in the forest. Whether it is due to an oversight (on the Pandava part) induced by wish to win back the kingdom, or whether it is a mistake of ours, should be told to us by Bhishma by calculating the shortness or excess…
Bhishma says (Mahabharata, Critical Edition, Virata Parva, Section 47, Stanzas 1 to 5)
The wheel of time revolves with its divisions – Kalas and Kasthas and Muhurtas and days and fortnights and months and constellations and planets and seasons and years. Because of their fractional excesses and the deviations, and also the deviations of the heavenly bodies, there is an increase of two months in every five years. I think that calculating keeping this in mind, there would be an excess of five months and twelve nights in thirteen years. Everything, therefore, that the Pandavas had promised, has been exactly fulfilled by them. Knowing this to be certain, Arjuna has made his appearance…
Many explanations have been given and many discussions held on the import of Bhishma’s words. He is clearly saying that theirs is a lunar calendar and every five years, two intercalary months are added. However, not many have fully explained the meaning of the five years and twelve days that he mentions when referring to the thirteen years of exile.
The following is my explanation for what the Mahabharata text says.
Their system of adding adhika masas (intercalary months) to bring lunar years in line with solar may have been simple: At the end of two years, they add one month, and at the end of five add another. So, two intercalary months in 5 years.
[The requirement is to add two months in five years. However, they may not have wanted to add one month in the middle of a year (two in five years, so the first after 2 1/2 years), as that would have made the reckoning of the months difficult. And, of course, rather than add two months at the end of five years, it made for more uniformity if they added the two months in stages. And hence the strategy of adding one month after 2 years and another after five. This system (one after two years and the other after five) of adding adhika masas is my speculation. This seems to be the only reasonable explanation for Bhishma’s words.]
Coming to the Pandavas, at the end of the thirteenth year, which would be in the third five-year cycle of the period, Bhishma knew that five months have been added to the calendar: 2 months in the first five-year cycle, another two in the next five-year cycle and one after the 12th year (following the logic above). He knew that to bring the lunar year in consonance with the solar at the end of the thirteenth year some number of days need to be added. He does a quick calculation (the Pandavas may have done the same calculation) and feels that 12 days needs to be added (2 months [=2*29.5=59 days] in 5 years. So, 59/5 = 11.8 [~12] days in one year).
(Actually, he needed to have added only six days [since the one month added after two years would take care of two-and-a-half years of the five year cycle] but rather than go with exact calculations, he went with looking at the calendar [whatever form it was in!] in front of him and added 12 days to the end of the 13th year to get the solar year completion of 13 years.)
The Pandavas also would have done the same calculations and added the 12 days. This gave them ample cushion (of six extra days) to make sure.
[If you want to be mathematically exact, Bhishma need not have added any extra days at all at the end, if they had followed the possible system that I had indicated. Number of solar days in 13 years = 13*365.25=4748.25 days; By adding the intercalary month after the second year and the fifth year, we get lunar days = 29.5*12 (year 1) + 29.5*13 (year 2) + 29.5*12 (year 3) + 29.5*12 (year 4) + 29.5*13 (year 5) + 29.5*12 (year 6) + 29.5*13 (year 7) + 29.5*12 (year 8) + 29.5*12 (year 9) + 29.5*13 (year 10) + 29.5*12 (year 11) + 29.5*13 (year 12) + 29.5*12 (year 13) = 354 + 383.5 + 354 + 354 + 383.5 + 354 + 383.5 + 354 + 354 + 383.5 + 354 + 383.5 +354 = 4749.5 days. So, by my (speculated) system, at the end of the 13th lunar year, they would have completed the 13th solar year with a day to spare. Note that this is valid only when an intercalary month is added after 2 years in a 5-year cycle! Of course, The Pandavas (and Bhishma) did not care to make such exact calculations. They just wanted to ensure that however you look at it (solar or lunar), 13 years had elapsed since the start of the exile.
Another thing to note is that this technique adapted by the people of that period, adding 2 months for every 5 years, gives an excess of 2.75 days in the lunar calendar over the solar. By solar calendar, number of days in 5 years = 365.25*5 = 1826.25 days; By lunar calendar, the number is 29.5*12*5 + 29.5*2 = 1829 days. So after around 55 years, they will have to drop one of the excess 2 months.]
Duryodhana assumed that when Arjuna appeared as himself at the end of the 13th year, some days were lacking to make it 13 solar years. But, Bhishma tells him about the 12 days after the lunar year end that the Pandavas had allowed to elapse before exposing their hidden identities.
There is another interesting episode that needs to be added to this story. Kichaka, the commander of Virata’s forces, had been lusting after Draupadi who was disguised as Virata’s queen’s maid. An upset Draupadi has Kichaka (and his kith and kin) killed by Bhima, and let it be known (so that Bhima’s identity is not exposed) that her husbands were Gandharvas and they had killed Kichaka in anger. The Virata king, worried that the kingdom may incur the wrath of the Gandharvas, asks his queen to let her maid go. But, Draupadi asks for thirteen more days of stay in the Virata kingdom, after which she would leave (Mahabharata, Critical Edition, Virata Parva, Section 23, Stanzas 24 to 28). So, it looks like Kichaka was killed on the last day of the 13th year by the lunar calendar. Draupadi knew that even though the 13th year was ending that day, there needs to be another 13 days (including that day) for the solar year to end, so that their exile is over by all calendars.
You can see that the confusion arising from having two calendars – solar and lunar – had started as long back as the Mahabharata period.