Solution to Zelda Puzzle
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We are given that Link does not have Triforce piece #3.  Hence he has not been into the third Labyrinth, which means he does not have that labyrinth's treasure, the raft.  Without the raft it is impossible to reach Labyrinth 4, so Link also lacks that piece of the Triforce and the treasure found there, the ladder.  Without the ladder one cannot finish Labyrinth 5, because a strip of water runs across one of the rooms there.  Link therefore lacks Triforce piece #5 and the treasure for that Labyrinth, the magic whistle.  In turn, the magic whistle is necessary for accessing Labyrinth 7.  Link therefore lacks piece #7 and the red candle, which is found in Labyrinth 7.  As we know he also lacks the blue candle, Link is unable to burn the tree that leads into Labyrinth 8.  That piece too, then, is not in his collection.

So much for what Link doesn't have.  What does he have?  We know that he has defeated a Gleok.  There are only three Gleoks in the first quest: in Labyrinths 4, 6 and 8.  Clearly the Gleok he defeated was the one in Labyrinth 6.  It follows that he has Triforce piece #6, and therefore has defeated the boss which guards it: a Gohma.  Now, the only way to kill a Gohma is with an arrow to the eye, so it follows that Link has the bow.  This is found in Labyrinth 1, so Link has that piece of the Triforce as well.  It remains only to determine whether he has piece #2 or not.

Clue #4 gives us four statements, only one of which is true.  Statement a) tells us that Link has gone north from the Lost Hills.  The Lost Hills are, according to the map included with the game, the room high in the northeast of the Overworld which repeats unless you walk west, or continually four times north.  Walking north from the Lost Hills leads to the entrance of Labyrinth 5, but as we know Link did not visit this screen, we know statement a) is false.  Statement b) tells us that Link has not gone west from the Lost Woods, which are the counterpart to the Lost Hills, a cross of repeating trees in the brown forest which can only be exited to the west after a secret sequence of moves is learned.  The key to resolving this statement's status is that we know Link has been to Labyrinth 6.  Labyrinth 6 is found in the northwest region of the Overworld which can only be entered in three ways: by crossing the river with a ladder, by using the power bracelet, or by passing west through the Lost Woods.  As we know that neither of the first two took place, we can deduce that Link has in fact gone west from the Lost Woods, and hence statement b) is false.  Statement c) tells us that Link has obtained a heart container from the extreme east edge of the Overworld.  There are two such heart containers; one can be reached only with the ladder, one with the raft.  Since Link has neither of these treasures, we know that this statement must be false as well.  It follows that statement d) is true, and therefore that Link has the magic boomerang.  This boomerang is found in Labyrinth 2, so he must have the second piece of the Triforce.

To summarize, then, Link has pieces number 1, 2 and 6 of the Triforce.

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