The note should rather be entitled The So-called Synoptic Problem. That is because the only problem that exists is that which men have invented. There is no synoptic problem.
Enough has been written about this subject to fill a large library. Every conceivable “solution” has been proposed, with different varieties of theories put forth. I will not go over even some of the most popular proposals. That would be a waste of time.
Remember (see my note entitled, Fact – Opinion – Faith), that when it comes to a subject like this, no one knows beyond any doubts the order and even the kinds of the source materials used. Added to that, significant argument (see below) can be raised against all of the long held views regarding the theory of which works were sources for the others. We are all (diligent seeker and scholarly expert alike) on the same footing. We all are left with educated opinion.
Here is my opinion. The synoptic problem vanishes when you take these facts into consideration.
Matthew was written first by the Apostle Matthew to the Jews to convince them that Jesus was the Messiah of Israel and the fulfillment of all of the Law, the Prophets, and the Writings of the Old Covenant Scriptures. He was the hope and complete salvation of Israel. The final document, which we have as canon, was most probably produced after some edits using oral tradition and notes that he used and wrote.
“Mark” was written after Matthew by the Apostle Andrew mainly to the gentile subjects of the Roman Empire to convince them that Jesus, and not Augustus, was the Savior of the world who brought the good news of salvation and deliverance from the judgment of God. What was said about Matthew is true about Andrew. The final document, which we have as canon, was most probably produced after some edits using oral tradition and notes that he used and wrote. Added to this, Andrew most likely constructed his own codex. At this point in time, single stitching was the technique used. Because of its inferior construction quality, it is highly likely that the last section of his codex got detached from the book very early in its existence. This accounts for the abrupt ending.
Luke was written after Matthew, and probably after Andrew. It was written by Luke the physician, and assistant to Paul, to a certain prominent man named Theophilus. Whether or not Theophilus was a pseudonym used to protect that individual from undue retaliation is not now known. The purpose of Luke was to convince Theophilus to be assured in the faith through a thorough and accurate narration among the many reports about the life and teachings of Jesus that were then current.
The issue that proves that one of these finalized narrations was not a direct model source for the others, which is the foundational argument of modern day synoptic problem studies, is the unassailable fact that the small and/or insignificant words (what I call personality words) of one writer do not appear in another. These are harmless words that do not impact the real meaning of any statements made. They are not offensive in any way. They do not reduce the meaning of the messages in any way. It is inconceivable not to maintain that if one writer used another directly as a source that at least a few of those words would appear in the secondary writer(s). They do not. This bedrock fact declares that each of the three writers did not use a canonical “gospel” as a primary source or even a secondary source.
The overlapping and repetition of narration and teaching that is seen among the three were due to the oral tradition and the abundant note-taking that both were prominent features in communication in the ancient world.