In Matthew’s gospel, Jesus is tempted by Satan in the wilderness in the following order:
- To turn stones into bread (Matt. 4:3-4),
- To jump off the temple (Matt. 4:5-7),
- To acquire the kingdoms of the world (Matt. 4:8-10).
However, in Luke’s gospel, the order is:
- To turn stones into bread (Luke 4:3-4),
- To acquire the kingdoms of the world (Luke 4:5-8),
- To jump off the temple (Luke 4:9-12).
Thus, they reverse the order of the 2nd and 3rd temptations.
A clue to harmonizing the two accounts may be found through examining the Greek wording. Namely, Matthew uses the more specific time-sequence Greek words tote (“then,” Matt. 4:5) and palin (“again,” Matt. 4:8) to introduce his temptations, whereas Luke uses the less specific kai (Luke 4:5), and, perhaps, de (“then,” Luke 4:9), to introduce them .
Additionally, several Old Latin witnesses, at least one Vulgate manuscript, and Ambrose’s Commentary on Luke rearrange Luke’s gospel to follow the order of Matthew’s (Metzger 2006, 114). This means that ancient witnesses also favored Matthew’s ordering as the correct ordering, and thus tried to modify Luke’s account.
Furthermore, Luke is known to record events out of chronological order for, presumably, theological reasons (e.g. Luke 3:18-21, where Luke records John’s imprisonment before Jesus was baptized by John).
Finally, and perhaps most significantly, the content of each temptation, and the themes of each writer, also both point to Matthew retaining the correct ordering.
Consider that 2 temptations begin with, “If You are the Son of God,” (Matt. 4:3 = Luke 4:3; Matt. 4:6 = Luke 4:9) while the other temptation says, “…if You will (fall down and) worship me,” (Matt. 4:9 = Luke 4:7). It seems (at least marginally) that the questions of Jesus being the Son of God would both come first, with the climactic temptation to worship Satan coming last. And this is precisely the order Matthew gives.
Further still, there is a clear theme running throughout Luke’s gospel that may influence the ordering he assigned, namely, Luke draws out Jesus traveling to Jerusalem as His final destination (cf. Luke 9:51, see also Reconciling Matthew’s and Luke’s Birth Narratives and Resurrection Harmonization).
Thus, it fits Luke well to thematically place Satan’s Jerusalem temple temptation as the final temptation that the others were moving toward (especially since Jerusalem’s temple is the very place Luke ends his entire gospel, cf. 24:53), even while he is careful to not record this as if it was the chronological sequence of events.
 I say “perhaps,” because some copies of Luke’s gospel have the word, “de” in Luke 4:9 (such as those used by the New American Standard Bible), while other copies instead have the word, “kai,” in Luke 4:9 (such as those used by the King James Version and New King James Version).
 It is unfortunate that the NKJV translation does not indicate this important difference when they translate Luke’s kai as, “then,” thus making it appear to English readers that Matthew and Luke contradict each other in introducing the temptations.
 I’m not endorsing their rearrangement of Scripture, nor am I arguing that Luke originally wrote the temptations in the same order as Matthew. Rather, I’m citing these ancient witnesses as added testimony that Matthew’s ordering, not Luke’s, was seen as the standard sequence.