2. Floor's class consists of 16 students, including Floor herself. All students took a test with four questions. Each question was worth a (positive) whole number of points. Each question was either completely correct or completely incorrect; no partial points were given. The question worth the most points was exactly 4 points more than the question worth the least points. All students achieved different scores; Floor herself got everything correct. How many points did Floor score at minimum?
B2. 21

There are 16 students and $2^{4}=16$ possibilities for which questions can be answered correctly. Since this is the same and everyone has a different score, each combination of questions must be worth a different number of points. In particular, each question must be worth a different number of points. We will now systematically try to determine how many points the question worth the least can be.

Suppose this question is worth 1 point. Then the question worth the most points is worth 5 points. There are at least 0 and at most $1+3+4+5=13$ points to be earned for the entire test, so there are at most 14 possible point totals. But we need at least 16 possible point totals, so this is not possible.

Suppose the question worth the least points is worth 2 points. Then the question worth the most points is worth 6 points. There are three possible options for the number of points per question: $2,3,4,6$ or $2,3,5,6$ or $2,4,5,6$. The second option is not possible, because the student who answered the questions worth 2 and 3 points correctly would have the same score as the student who only answered the question worth 5 points correctly. For similar reasons, the first and last options are not possible: then the student who answered the questions worth 2 and 4 points correctly would have the same score as the student who only answered the question worth 6 points correctly.

Therefore, we consider what happens if the question worth the least points is worth 3 points, and the question worth the most points is worth 7 points. There cannot be a question worth 4 points, because then the student who answered the questions worth 3 and 4 points correctly would have the same score as the student who only answered the question worth 7 points correctly. So the four questions are worth $3,5,6$ and 7 points, and we can verify that all possible combinations of zero, one, two, three, and four questions answered correctly yield different point totals. Floor therefore had at least $3+5+6+7=21$ points on the test.