Argument 1 is clearly wrong.
Consider the island with only two blue-eyed people. The foreigner arrives and announces "how unusual it is to see another blue-eyed person like myself in this region of the world." The induction argument is now simple, and proceeds for only two steps; on the second day both islanders commit suicide. (I leave this as a crucial exercise for the reader.)
Now, what did the foreigner tell the islanders that they did not already know? Say that the blue-eyed islanders are $A$ and $B$. Each already knows that there are blue-eyed islanders, so this is not what they have learned from the foreigner. Each knows that there are blue-eyed islanders, but neither one knows that the other knows this. But when $A$ hears the foreigner announce the existence of blue-eyed islanders, he gains new knowledge: he now knows that $B$ knows that there are blue-eyed islanders. This is new; $A$ did not know this before the announcement. The information learned by $B$ is the same, but mutatis mutandis.
Analogously, in the case that there are three blue-eyed islanders, none learns from the foreigner that there are blue-eyed islanders; all three already knew this. And none learns from the foreigner that other islanders knew there were blue-eyed islanders; all three knew this as well. But each of the three does learn something new, namely that all the islanders now know that (all the islanders know that there are blue-eyed islanders). They did not know this before, and this new information makes the difference.
Apply this process 100 times and you will understand what new knowledge was gained by the hundred blue-eyed islanders in the puzzle.
$3008 = 64 \times 47$, which is the only decomposition that makes sense here ($47$ being prime). Each piece will then be, on average, $1.79\mathrm{cm} \times 1.75\mathrm{cm}$.
You should have $4$ corner pieces and $214$ edge pieces.
Best Answer
The answer is in the title of this website - Stack. Measure the thickness of one piece, then stack them up and measure the height, place them in a tube, preferably clear, of small enough diameter so the pieces won't fall down the side. A graduated cylinder would be perfect.