(a) How many students must be in a room to be sure that at least two are the same sex? (b) How many students must be in a room to be sure that at least nine are boys or at least nine are girls? (a) There must be at least \( \square \) students in a room to ensure there are at least 2 of the same sex. (b) There must be at least \( \square \) students in a room to ensure there are at least nine boys or nine girls.

**(a)** We consider the two sexes as two groups (boys and girls). In the worst-case scenario, if there are only two students, they could be of different sexes—one boy and one girl. To ensure that at least two students belong to the same sex, we must add one more student. Thus, by the pigeonhole principle, having \[ 3 \] students guarantees that at least two are of the same sex. **(b)** We want to guarantee that there are at least nine boys or at least nine girls. Consider the worst-case scenario where the numbers are as balanced as possible but still avoid having nine of either: the maximum number of each without reaching nine is \(8\). With \(8\) boys and \(8\) girls, there are \[ 8 + 8 = 16 \] students, and neither group has reached nine. Therefore, adding one more student (making a total of \(17\) students) forces one group to have at least nine students. Thus, we need \[ 17 \] students to ensure there are at least nine boys or at least nine girls. **Answers:** (a) There must be at least \(\boxed{3}\) students in a room to ensure there are at least 2 of the same sex. (b) There must be at least \(\boxed{17}\) students in a room to ensure there are at least nine boys or nine girls.