How do we determine the nature of the reign of the kings when the numbers don't seem to add up?

Question

In the book of 2 Kings, the kingdoms of Judah and Israel were separate.

In Amaziah's (Judah) 15th year, Jeroboam (Israel) began to reign. In Jeroboam's (Israel) 27th year, Azariah (Judah) began to reign. That is, Azariah (Judah) began to reign 15+27=42 years after Amaziah (Judah) began to reign. However, Amaziah (Judah) only reigned 29 years. I've tried to illustrate this below:

ISRAEL:           Jeroboam (reigned from the 15th year of Amaziah)
JUDAH: Amaziah (reigned 29y)  -->   Azariah (reigned from the 27th year of Jeroboam)

(see Melachim 2 / 2 Kings 14:2, 14:23, 15:1)

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Similarly, Ahaz began to reign in Pekah's 17th year. Hoshea began to reign in Ahaz' 12th year. But Pekah only reigned 20 years (leaving a gap of 9 years).

ISRAEL: Pekah (reigned 20y)  -->  Hoshea (began to reign in Ahaz' 12th year)
JUDAH:            Ahaz (began to reign in Pekah's 17th year)

(see Melachim 2 / 2Kings 15:17, 16:1, 17:1)

I know that many have tried to make the chronologies fit by including co-regencies (two kings reigning at the same time). But the word מָלַךְ is the same for all of these kings - wouldn't all of the מָלַךְs either refer to the beginning of a co-regency, or the beginning of a reign alone? And if not: how do we differentiate the beginning of a co-regency from the beginning of a reign alone, based on the words used or the context?

Answer 1

It is important to remember that Melochim was not written as a history book, which is why it actually constantly says "and the rest of King So-and-so's matters are written in the histories of the Kings of Yehuda/Yisrael". Instead, it is extracting the parts of history needed to show that the kings who were righteous were successful, and the kings who were not righteous were not successful.

So the word Malach can be used to mean ruled in different capacities, because the people with the histories would know what the author meant.

Answer 2

But the word מָלַךְ is the same for all of these kings - wouldn't all of the מָלַךְs either refer to the beginning of a co-regency, or the beginning of a reign alone? And if not: how do we differentiate the beginning of a co-regency from the beginning of a reign alone, based on the words used or the context?

I am not sure it is really possible to understand the nature of the reign of each king merely from the context in close vicinity to the verse(s) referring to the start and end of the reign. One of the major issues in studying Tanach is trying to understand things from the POV of the writers and their original audience (we, later readers, are a secondary audience). In other words, since the writers (typically prophets) lived in a certain time period and were directing their words at certain people, there were naturally things left unsaid, or at least, not clearly emphasized, because all of these people already knew all of these things. There are things that were accepted as social conventions, and therefore there was no need to state them.

For example, there is a thesis in Hebrew that I have read part of, called "Reconstruction of the Chronicle Used by the Author of the Book of Kings in Light of the Mesopotamian Chronicles", by Dr. Yosi Baruchi. The thesis attempts to reconstruct some of the bibliography that the author of Sefer Melachim may have based himself on. Post-reconstruction of the sources, Dr. Baruchi attempted to reconstruct the chronology of the list of kings. As he himself stated, the differences between the MT (Masoretic Text) and the LXX point to (somewhat) bad attempts by the LXX translator(s) to correct the perceived chronological issues in the book (p. 151 and n. 1). However, in order to properly understand the MT's chronology, there are a few assumptions that must be accepted:

1. The years of the kings, both of Yisrael and of Yehudah were not counted from the start of the reign but from Rosh Hashanah [La'melachim] (the New Year [of the Kings])¹.
2. Rosh Hashanah in the two kingdoms was not at the same time: For the kings of Yehudah it was in the spring while for the kings of Yisrael it was in the fall.
3. Another key difference between the two kingdoms: The time that passed between the crowning of a king of Yehudah until the first Rosh Hashanah was counted as part of the reign of the previous king, while in the kingdom of Yisrael it was counted as part of the reign of the new king. (pp. 151-152 and related notes)

These are rules that have been concluded from analyzing the lists in their entirety, as well as perhaps other portions of this book (consider, for example, Melachim 1:12:32-33 which shows that Yerov'am changed the calendar of the kingdom of Yisrael).

Even if these rules are incorrect or not entirely correct, I believe they give us the understanding that there were rules, and very likely there were different rules in each kingdom, and whoever lived at the time likely knew these rules and did not need to have them included in the book. We find, therefore, that behind the term "מָלַךְ" were all of these different factors that defined the length of the reign, but since it was obvious that were differences between reigns depending on various factors, there was no need to write anything other than "מָלַךְ".

I hope that clarifies things.

As a side-note, I found today in the library an edition of פירוש רבנו מיוחס ב"ר אליהו on Chronicles and there he states (at least once; I haven't gone through the entire commentary yet, short as it may be) on Chronicles 2:16:1 (emphasis mine):

"שלשים ושש למלכות אסא. אי אפשר לומר כן שהרי בעשא מת בשנת עשרים ושש לאסא כמו שמפורש במלכים ולא נכתב אלא לדורשה. וכן דרשו רבותינו כנגד שלושים ושש שנה שנתחתן שלמה לבת פרעה כנגדן נגזרה גזירה על מלכות בית דוד לבטל ועוד תחזור להם וכנגדן שלשים ושש שנה שנגזרה על מלכות ארם שתמשול בישראל ועוד תפול ביד בית דוד."

In other words, in his view, whenever there is a chronological discrepancy, that means that the number was not intended to be understood simply but as something symbolic. Perhaps other commentators state this as well. I have yet to check.

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¹ He only refers to it as Rosh Hashanah, but I added "La'melachim" to differentiate between it and the "regular" Rosh Hashanah.

Answer 3

If necessary I would respond to all issues about the Kings' chronology. Otherwise I will refer below to sources on the topic.

Seder HaDoros

Malbim

See Seder Olam with Hagahos of Yaavetz and Vilna Gaon.

Aderes Eliyahu (Vilna Gaon) with table of Judges and Kings. http://www.hebrewbooks.org/pdfpager.aspx?req=26873&st=&pgnum=136

Keep in mind: All king lists of all histories assume some overlap due to different kings beginning and ending reigns on different dates as well as some kings having a different reckoning for when they officially assumed the throne as opposed to practically, actually doing so...