Follow up to Product & Sum.
As in the “three wise men” puzzle, it’s useful to begin by analyzing simplified variants of the problem. Lets start analyzing the variant where only one islander has blue eyes and the other 999 have brown eyes.
When the foreigner gives his message, as the blue-eyed islander can see that the other islanders are all brown-eyed, he now knows his own eye color and must commit suicide the following day at noon. But, when the blue-eyed islander kills himself, all the other islanders know that they are brown-eyed and they must commit suicide at noon the next day (we can assume that each islander knows that his eye color is either blue or brown).
Now we can examine the variant where two of the 1000 islanders are blue-eyed. In this case the foreigner’s message does not give any information immediately, as any inhabitant of the island knows by observation that there is at least one blue-eyed islander. But the absence of a suicide at noon the following day shows to each of the two blue-eyed islanders that the other one is not the only blue-eyed inhabitant of the island. Then, in the second noon after the message, the blue-eyed islanders commit suicide, followed by the rest of the tribe the next day.
This reasoning can be extended to the full case, where the 100 blue-eyed islanders will commit suicide at the 100th noon since the foreigner’s message and the 900 brown-eyed ones will do the same at the following noon (the 101st).