According to these Google Ngrams, both American and British English use exactly the same more than the exact same. Here is the usage in American English:
http://ngrams.googlelabs.com/chart?content=the%20exact%20same%2Cexactly%20the%20same&corpus=5&smoothing=3&year_start=1800&year_end=2000
And here it is in British English:
http://ngrams.googlelabs.com/chart?content=the%20exact%20same%2Cexactly%20the%20same&corpus=6&smoothing=3&year_start=1800&year_end=2000
Despite its usage, the exact same is considered informal (but is not deemed incorrect) by this site at Washington State University:
In casual speech we often say things like, “The fruitcake he gave me was the exact same one I’d given him last Christmas,” but in formal English the phrase is “exactly the same.”
However, there is a long discussion of the phrase which writes that:
The traditional construction is “exactly the same time,” with an adverb (“exactly”) properly modifying an adjective (“same”).
Critics of a phrase like “the exact same time” condemn it because “exact” (an adjective) is being used as an adverb (like “very”)....
Proponents of the phrase note, however:
Elsewhere, the Cambridge Grammar notes that noun phrases including “the same” often include modifiers to reflect varying degrees of sameness. Sometime modifiers come after “the” (as in “the very same mistake”), and sometimes before, as with “much,” “almost,” “roughly,” and “exactly.”
I would add “exact” to the list of modifiers that can follow “the” (as in “the exact same mistake”). In my opinion, this usage is acceptable in all but the most formal writing.
If you’d like another authority, The Columbia Guide to Standard American English also says “exact same” is “standard in all but the most formal and oratorical contexts.”
Both phrases are redundant, and the exact same can be considered correct or not depending on which style guide one subscribes to. An American English grammar guide specifically mentions that the phrase is mostly standard, and a British English guide notes that there are similar phrases. So use depends on context: in formal writing, avoid it, but in anything else the exact same can be (again, it depends on who you follow) acceptable.
Rarely, must is used as a past tense. Belshazzar, by H. Rider Haggard, has we went because we must, in a prose style which is perhaps deliberately archaic to reflect the ancient Egyptian context.
In this odd snippet, If Thoreau went because he would, Hawthorne went because he must, one might say the author is "playing with language".
But here's Ralph Waldo Emerson with What he did, he did because he must. I would not wish to say Emerson doesn't know his own language.
From comments under @Henry's answer, it seems something quite odd has been going on. Many people will know the archaic present tense mote because Freemasons & such still say So mote it be in a "ritual" context. Bizarrely, the past tense "must" eclipsed "mote" for present tense usage. But in so doing, "must" somehow almost completely lost its ability to still be used as a past tense.
In spite of all the above, ordinary mortals in ordinary contexts today should stick with the standard position put forward by other answers. Use had to for the past tense.
Best Answer
"Exact same" represents a grammatical practice that is particularly prevalent in American English; the use of an adjective for an adverb. In this phrase "exact" modifies "same" and is functioning as and adverb.
In the literal sense "exact same" is indeed redundant, however, words aren't quite so precisely defined as apparently your teach would have you believe. If I have a Hugo Bos blue shirt with an 18 inch collar, someone with a Hugo Bos blue shirt with a 20 inch collar might think we have the same shirt. In fact, someone with a Hugo Bos white shirt with a 20 inch collar might think we have the same.
You might even argue that if they are two shirts identical in every respect they are still not "the same" shirt. If I wore my shirt today, and again tomorrow, you might tell me "you're wearing the same shirt as yesterday", and that would be absolutely literally true. Which is to say, "same" is used rather more loosely than "the identical object" in common language.
By modifying it with "exact" you are emphasizing that they are even more "same" than if you did not so modify.
Many words sound like they are absolute, binary, and not subject to gradation. However, I am reminded of a discussion between Sheldon Cooper and Stuart the comic book guy on the hilarious TV show "The Big Bang Theory":