Claimants to this supposed contradiction really only show one thing: they haven't actually bothered to read any of the contextual information surrounding these passages. This is shown because, as we'll see below, this contextual data demonstrates that Jesus foretold His death to His disciples more than once.
In the account of Luke, we note that the timeline of events leading up to his transfiguration account is abbreviated compared to that of Matthew.
In the account given by Matthew, we see a fuller revelation of the events which occurred during this period in time, and they show us that the events given in Luke happened around two days before the time of the statement of Christ which is being questioned in Matthew. They record thus:
Note, the asterisk denotes the ONLY point of congruity between these two accounts. This point is the feeding of the five thousand, which occurred about eight days before the transfiguration.
The crux of the sceptic's claim for a contradiction between these two accounts is the assumption that the accounts of Jesus' foretelling His coming death, et. al. each refer to the same single discourse to His disciples. This is not a safe assumption to make, however. It is certainly reasonable to believe that there was more than one single time in which Jesus told His disciples these things. After all, we see examples of repetition or near-repetition of the same themes of Jesus' preaching at other points in the Bible. The most obvious example of this would be the basic message of the Sermon on the Mount (The Beatitudes - Matthew 5:2-16) which was reiterated and added to by Jesus in His "Sermon on the Plain" (Luke 6:20-36).
Further, we see that Jesus' questioning of His disciples in Matthew, said to occur near Caesaria Philippi, not Bethsaida, was not the same event as that recorded in Luke. This is shown in that Jesus, at the time in which He fed the four thousand (after the five thousand, remember), said that the multitudes had continued with Him "three days" (Matthew 15:32). We can reasonably garner from this that this crowd had been with Him since the first feeding, two days earlier, and were now spending their third day with Him (wouldn't most people probably stick around with someone who could generate large quantities of food from nothing, at least for a little while?) Hence, the feeding of the FOUR thousand was two days after the feeding of the FIVE thousand.
Further, Matthew tells us that Peter declared Jesus to be the Christ, and was commended, and that AFTER this point, "from that time forth began Jesus to shew unto his disciples" (Matthew 16:21) that He would be taken to Jerusalem, killed, and would rise again. This phrase "from that time forth" indicates passage of time. Peter's gallant but wrongheaded contradiction of Jesus on these matters occurred AFTER Jesus had been discoursing for some time, perhaps after quite a long sermon, which indicates a different conversation than the one presented in Luke, which does not depict Peter challenging the Lord's prediction. The timeline indicates that Peter rebuked the Lord around two days after the initial teaching on by Jesus concerning His death (Matthew 16:22), which indicates that Peter had been using the time since the initial presentation to mull over the concept. We know from the Gospel presentation that Peter didn't really catch on to the real import of the Lord's message until after the events of the death and resurrection (though he certainly had zeal before that time!) The intervening time between the Mattean and Lukan sermons were probably used by Peter to meditate on what Christ had said earlier and subsequently repeated in Matthew 16.
Also, we see that the charactre of Jesus' discourse to His disciples in Matthew is a bit different than the one recorded in Luke which occurred two days earlier. In the Mattean account, Jesus used the opportunity to lay down some foundational truths about the building of His church and the responsibility of a follower of Christ to take up his or her cross and follow Christ, that is to say, to be willing to give full devotion to Christ alone. These topics, very important, were not discoursed on in Luke's account, and hence, indicate that these two conversations are not the same event, and that Jesus was building upon His earlier teaching which was presented in Luke.
Would Jesus have repeated His question to His disciples? Wouldn't an answer from them one time be enough? The Bible presents other examples of Jesus using repetition in His questions to prove the words of the one who was answering Him. This He did with Peter when He questioned Peter three times concerning his devotion to Christ (John 21:15-17). Thus, there's no real room to discount the repetition of Jesus' discourse to His disciples on the matter of His coming death and resurrection, especially as that was to be a foundational set of doctrines for the church which Christ was building. Nothing aids learning like repetition, as they say.
Hence, we see from both the chain of events recorded in the Gospel accounts, and from the context of the passages, that the two accounts in which Jesus told His disciples about His death and resurrection, and told them that some of them would not die before they saw Him coming in His kingdom (an allusion to the transfiguration) are not the same single event. Rather, Luke's account did indeed occur about eight days before the transfiguration, while Matthew's account (with all that intervening travel and disputation with the Pharisees and whatnot) of the conversation logically did indeed occur SIX days before the transfiguration.