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Once upon a time, a Sultan captured wise men and kept them in separate rooms. In the Sultan's hall stands a glass, which is currently standing bottom down. Every minute the Sultan calls one of the wise men, chosen at random, to the hall; the man who is called may either turn the glass over or leave it as it is. The calls go on forever, so every wise man will be called again and again, infinitely many times. The wise men are freed only if one of them correctly declares that all of them have been called to the Sultan at least once. If the declaration is wrong, everyone is put to death. Before being locked in their separate rooms, the wise men were allowed to confer only once.
Does there exist a strategy that guarantees that the declaration, when made, is correct (and that it is eventually made)? If yes, what can it be? If no, prove why.