Question: The prophet Daniel incorrectly states that in the third year of the reign of King Jehoiakim Nebuchadnezzar is king and that he conquers Judah.
The third year of Jehoiakim's reign was 606 BCE, at which time Nebuchadnezzar was not yet king of Babylon. It was in 597 BCE that Nebuchadnezzar takes Jerusalem, by then Jehoiakim had died.
(Dan 1:1 NRSV) In the third year of the reign of King Jehoiakim of Judah, King Nebuchadnezzar of Babylon came to Jerusalem and besieged it.
(Dan 1:2 NRSV) The Lord let King Jehoiakim of Judah fall into his power, as well as some of the vessels of the house of God. These he brought to the land of Shinar, and placed the vessels in the treasury of his gods."
Response: The third year of Jehoiakim's reign would have been 606 BC, if you use the Judah method of counting the years of a king's reign. But, if you use the Babylonian method, then the third year of Jehoiakim would have been 605 BC.
The Babylonians included an "accession year." When a king died during the course of a calendar year, that whole year, from the first day to the last day, was counted as the last year of that king's reign. As for the new king, that year would be regarded as the accession year. And, the counting of the years of the new king's reign would begin on first day of the next year.
Daniel lived in Babylon. And it would seem natural to me that he would use the Babylonian method of tracking time. It would be as natural as adjusting your watch if you moved to another country with a different Time Zone.
It is true that Babylon besieged Jerusalem in 597 BC, and, for that matter, again in 586 BC. But it is also true that Daniel, chapter 1, says that Babylon besieged Jerusalem in 605 BC. According to the Bible, Babylon besieged Jerusalem three times.
In Daniel, chapter 1, you'll see that there is no mention bloodshed or destruction. I just wanted to point that out because I suspect that some people overlook the 605 BC besieging because there is no mention in Daniel 1:1 of bloodshed and destruction. But, technically speaking, a besieging can include a simple show of force, without the use of force.
A quick side note: When Daniel 1:1 speaks of the third year of the reign of King Jehoiakim, he doesn't actually mention the year of Nebuchadnezzar's reign. I suppose that it is possible that Daniel could have referred to Nebuchadnezzar as King Nebuchadnezzar even if Nebuchadnezzar wasn't actually king just yet. I'm not saying that this is what Daniel did, I am just saying that it wouldn't necessarily matter if he did. It would be like saying, "I first met President Reagan when he was campaigning for his first term in office," even though he wasn't actually president yet.
A note about ancient dates:
1. It is not unusual for historical sources to differ by a year or two in the dates that they assign to various ancient events. Some historical sources say that the first year of Jehoiakim's reign as king of Judah began in 609 BC. Others say 608 BC. I've even seen one source that said "609/608 B.C." Same goes for the destruction of the first Temple - some say it happened in 586 BC and others say 587 BC.
2. The ancient Babylonian calendar did not begin January 1 and end on December 31. Neither does the Jewish civil calendar or the Jewish sacred calendar. That means that an ancient Babylonian year, or Jewish civil year, or Jewish sacred year, would overlap portions of two Gregorian calendar years (the calendar that we are using). And, a Jewish civil year would overlap portions of two Babylonian years. And, vice versa.