What is the synoptic problem? Much of the text of Mk, Mt, and Lk is very similar, and in some places identical, in either two or all three of these 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 suggested solutions generally involve copying and/or editing between Mk, Mt, and Lk, 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 Mk was the first of the synoptics to be written (this is known as Markan Priority), with Mt and Lk following, probably in that order.
There is a group of several possible solutions that fit this general pattern, and within this group the existence of the parallel material in Mt and Lk is usually accounted for by the aMt (the author of Mt) knowing Lk, aLk knowing Mt, or through the use of a hypothetical document (generally known as Q), known to both aMt and aLk, and being the source at least of some of their common material (The details of these and other possible solutions can all be found on the Synoptic Problem Website). The use of Q solves an otherwise difficult part of the problem, which is that although in the places where Mt and Lk have parallel material, Mt sometimes appears to be earlier (more primitive) than Lk, in other places Lk appears to be more primitive than Mt. This is referred to as ‘alternating primitivity,’ and is described in Fallacies at the Heart of Q by Mark Goodacre.
Because it is impossible for Mt and Lk 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 Mt or Lk 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, at the time of writing (Feb 2014) 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 the stylometric analysis described here is an example. The Conclusions of this analysis show strong support for Markan Priority, but also support for the view that some words common to Mt and Lk (but not Mk) came from another source, although this source does not appear to be what is commonly referred to as 'Q.' Instead, this source can be viewed as an early version of Lk, as described in The MwEL Theory: A New Synoptic Solution.
Also see Marcion's Gospel and the Synoptic Problem for information regarding Marcion's Gospel, and how it relates to Luke and the other synoptic gospels.