Deductions from Fact 1 (4 children) and Fact 3 (half are girls) prove that there must be 2 boys. Statement I (at least one girl has blue eyes) and Statement III (the boys have brown eyes) cannot be definitively proven true or false based on the given facts about the total number of blue/brown eyed children (Fact 2), as the distribution of eye color among boys and girls is not specified.