What is the synoptic problem? Much of the text of Mark, Matthew, and Luke is very similar to, and in some places identical to, text in either one or both of the other two gospels, and for this reason they are collectively referred to as the Synoptic (or ‘seeing together’) Gospels, or just the ‘synoptics.’ The existence of these synoptic parallels raises the issue of how they came to exist, and a very good general statement of the problem (usually referred to as the synoptic problem), with suggested solutions, is given by Stephen C. Carlson on the Synoptic Problem Website:
The synoptic problem is an investigation into the existence and nature of the literary interrelationship among the first three "synoptic" gospels. Matthew, Mark, and Luke are called the synoptic gospels, in contrast with John, because they can readily be arranged in a three-column harmony called a "synopsis." Unlike John, the synoptic gospels share a great number of parallel accounts and parables, arranged in mostly the same order, and told with many of the same words. Any proposed solution to the synoptic problem, therefore, must account for these literary similarities among the synoptics, not so much in terms of their factual content, but in the selection of that content, the arrangement of the material, and wording of the parallels.
It is rare for two independent reporters of the same event to share more than a few words in common, but the synoptic gospels often feature a substantial number of agreements in their exact words. For example, in one passage about John the Baptist, Matthew and Luke agree for 61 out of 63 Greek words of a presumably Aramaic speech. Generally, the verbatim agreement between Matthew, Mark, and Luke runs about 50% of the words, but, by contrast, their agreement with John in parallel episodes falls to about 10%.
Because of these agreements, a number of questions come to mind: How did the agreements arise? Were the words in one gospel copied from another, or did the common words come from other (unknown) sources? Where words are the same in two or all three gospels, can we tell the directionality, i.e. which was written first? The various hypotheses generally involve copying and/or editing between Mark, Matthew, and Luke, possibly supplemented by additional hypothetical source material, typically either earlier material from which one or more of the synoptics was derived, or a document containing material common to two of them. Although the synoptic problem is not ‘solved,’ the current majority opinion is that Mark was the first of the synoptics to be written (this is known as Markan Priority), with Matthew and Luke following, probably in that order.
There is a group of several hypotheses that fit this general pattern, and within this group the existence of the parallel material in Matthew and Luke is usually accounted for by aMatthew (the author of Matthew) knowing Luke, aLuke knowing Matthew, or through the use of a hypothetical document (generally known as Q, see What Exactly is Q) known to both aMatthew and aLuke, and being the source at least of some of their common material (The details of these and other hypotheses can all be found on the Synoptic Problem Website). The use of Q as a source solves an otherwise difficult part of the problem, which is that in the places where Matthew and Luke have parallel material Matthew sometimes appears to be earlier (more primitive) than Luke, but in other places Luke appears to be more primitive than Matthew. This is often referred to as ‘alternating primitivity,’ and is described in Fallacies at the Heart of Q by Mark Goodacre.
Because it is impossible for both Matthew and Luke to be earlier than each other, an additional document on which both depend is usually posited, either something like Q (although it should be noted that Q is a very slippery beast, with it’s content varying considerably depending on who is defining it, as indicated in Has Goulder Sunk Q? by Allan J. McNicol), or sometimes an earlier version of either Matthew or Luke that was also known to the author of the other gospel. If such an earlier version did exist, and it could be shown that it accounted for the alternating primitivity, then the need for a document like Q would disappear. However, to date (Dec 2015) it has not been decisively determined whether Q or some other document is actually required to 'solve' the synoptic problem.
One way of looking at the problem is to look at the style of text used in different parts of the gospels, to see (for example) whether the style of the text in an agreement matches that in other sections of the gospels in which the agreements occur. This is termed stylometic analysis, of which this stylometric analysis is an example. The Conclusions of this stylometric analysis show strong support for Markan Priority, but also support for the view that some words common to Matthew and Luke (but not Mark) came from another source, although this source does not appear to be what is commonly referred to as 'Q.' Instead, this source can possibly be viewed as an early version of Luke, as described in MwEL: A New Synoptic Hypothesis.
See also Marcion's Gospel and the Synoptic Problem for information regarding Marcion's Gospel, and how it relates to Luke and the other synoptic gospels.
Bigg, Howard C: The Present State of the Q Hypothesis, 1988
Bird, Michael F: The Holtzmann-Gundry Solution to the Synoptic Problem (Three Source Hypothesis). Also Goodacre, Mark: Mike Bird on Luke's use of Matthew and Q
Burkett, Delbert Royce: Rethinking the Gospel Sources: Volume 2: The unity or plurality of Q, SBL, 2009
Carlson, Stephen C: The Synoptic Problem Website
Derrenbacker, Robert A, Jr and Kloppenborg Verbin, John S: Self-Contradiction in the IQP? A Reply to Michael Goulder, 2001
Early Christian Writings: The Existence of Q
Farmer, William R: The Present State Of The Synoptic Problem, 1998
Foster, Paul: Is It Possible to Dispense with Q? , 2003
Gentile, David: A statistical approach to the synoptic problem
Goodacre, Mark: The Case Against Q: Studies in Markan Priority and the Synoptic Problem and The synoptic problem: a way through the maze. Online: The Case Against Q, Ten Reasons to Question Q, A Monopoly on Marcan Priority? Fallacies at the Heart of Q. See also Mark Q Overlaps I: Terminology, II: Major Agreements Between Matthew and Luke, III: Minor Agreements between Mark and Luke, IV: Back to the Continuum, V: the degree of verbatim agreement, VI: The Direction of Dependence
Goulder, Michael: Luke: A New Paradigm
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Just, Felix: The Synoptic Problem
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Kloppenborg, John S: On Dispensing with Q?: Goodacre on the Relation of Luke to Matthew, 2003
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Smith, Barry D: The Synoptic Problem, Crandall University
Streeter, B.H: The Four Gospels: A Study of Origins. Treating of the Manuscript Tradition, Sources, Authorship, and Dates
Text Excavation.com: The Synoptic Problem
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Waltz, Robert: The Encyclopedia of New Testament Criticism (online) or in PDF form
West, H. Philip Jr.: A Primitive Version of Luke in the Composition of Matthew, 1967
Wikipedia.com: Synoptic Gospels