On the chronology of the kingdoms of Israel and Judah
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A whole mass of difficulties appear for anyone who is trying to puzzle out the chronological details involved in the period in which the Northern Kingdom of Israel and the Kingdom of Judah existed simultaneously.

After the death of Solomon, Rehoboam inherits the throne of the united Israel, which quickly separates into the kingdoms of Israel and Judah. Each kingdom has its own kings, the lengths of whose reigns are given in the biblical text. The two kingdoms exist in parallel until the destruction of the Northern Kingdom. An interesting problem emerges if you try to measure the time from the split in the kingdoms till the demise of the Northern Kingdom. Let us try two methods: first the time-span in terms of the reigns of Southern kings, followed by the North.

A Southern chronology
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The split in kingdoms occurs at the beginning of the reign of Rehoboam. The length of reign of each of the Judahite kings is given on a another page, where you can find the relevant biblical citations. Rehoboam ruled 17 years, Abijah 3, Asa 41, Jehoshaphat 25, Jehoram 8, Ahaziah 1, Athaliah 7, Joash 40, Amaziah 29, Uzziah 52, Jotham 16, and Ahaz 16. So, from the beginning of the reign of Rehoboam to the end of the reign of Ahaz comes to a total of 255 years. Ahaz is succeeded by Hezekiah.

"And at the end of three years they took it: in the sixth year of Hezekiah, which was the ninth year of Hoshea king of Israel, Samaria was taken" (2 Kings 18:10). So adding six years of Hezekiah's reign, we get 261 years in which the two kingdoms existed in parallel.

A Northern chronology
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Now let us add up the reigns of the northern kings. They are Jeroboam (22 years), Nadab (2 years), Baasha (24 years), Elah (2 years), Zimri (7 days), Omri (12 years), Ahab (22 years), Ahaziah (2 years), Jehoram (12 years), Jehu (28 years), Jehoahaz (17 years), Joash (16 years), Jeroboam II (41 years), Zechariah (6 months), Shallum (1 month), Menahem (10), Pekahiah (2), Pekah (20), and Hoshea (9).

The relevant dates can be found in 1 Kings 14:20; 15:25, 33; 16:8, 15, 23, 29; 22:51; and 2 Kings 3:1; 10:36; 13:1, 10; 14:23; 15:8, 13, 17, 23, 27; 17:1

These reigns add up to 241 years, 7 months, 7 days. Compared to the 261 years we would get from looking at the "Southern" years, we've now got a discrepancy of 19 or 20 years.

What to do?
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This discrepancy was found by adding up figures. To add up figures, a number of assumptions must be made, including, most importantly, the assumption that each king's reign begins at the moment his predecessor dies. If we relax that assumption, the situation changes.

One way the situation changes is if we allow the possibility that the reigns of a king and his successor might overlap. According to Thiele's Mysterious Numbers of the Hebrew Kings, the reigns of Asa and Jehoshaphat overlapped, as did the reigns of Jehoshaphat and Jehoram, Amaziah and Azariah, Azariah and Joatham, and Jotham and Ahaz. And that's just the kingdom of Judah. In Israel, the reigns of Jeroboam and Zechariah overlap.

It is plausible for reigns to overlap, and there are some indications in the Hebrew Bible that some overlaps did occur. Other overlaps are strictly hypothetical -- they are simply assumed by Thiele on the grounds that they make discrepancies disappear.

So it is hard to know for certain where and when co-regencies may have occurred. Certainly, we can assume them whenever we wish to make a discrepancy vanish, but then this seems to rest on the assumption that errors in the chronological data are impossible. But, of course, errors in the Hebrew Bible's chronological data are entirely possible, unless we are willing to accept the claim that Saul became king at one year old and reigned two years over Judah.

So we may never come to a complete chronological solution to the problems of the Hebrew Bible. Nevertheless, regardless of the internal problems with the chronological data given in Kings, there are enough references in the Hebrew Bible to known historical events, and enough references to biblical kings in extrabiblical sources, something approaching a timeline of the biblical kings is agreed upon. Whether you are consulting the timelines of Albright, Thiele, Galil, or Kitchen, they all seem to line up within a few years for most kings, with the greater areas of uncertainty stretching to one or two decades at most. There is more fuzziness around the tenth and eleventh centuries than in the ninth and later.