# What a weird final exam

someday, some math teacher decided to test the logic deduction capacity of the class consists of 32 students. When the class began, he showed a letter to each student between A to E and said:

"There will be the final exam in an hour in the middle of the semester, there will be only answer sheet, this is a pop up Final!

I actually gave you one of multiple choice’s answer to each one of you, from question 1 to 32 by the give away order. i am generous, so the exam will be out of 128 and every right answer has same point.

I will give you a chance to communicate with at most 3 people at the board and you can share your answers. No cheating more than that is allowed! no talk, no chat, nothing else but taking notes from others.

Every time you come together and share notes, you will come to the board and take notes from each other only and I will count how many times you gather up. and everybody will lose $$1$$ point every time you gather.

now you have an hour! go!”

What is the maximum average point they can get as a class?

and

What is the highest point they can get if every student wants to know all answers?

Fyi: Students want to share what they know all the time.

• May I request a couple of clarifications? (1) When they take the exam, are they allowed to guess for questions they don't know? (2) When groups of students confer, are they allowed to pass on what they've learned from one another or can they only share the letters they were given at the outset? (My guess is that the intended answers are "no" and "they can pass on what they've learned".) – Gareth McCaughan Aug 11 '18 at 16:38
• 1- they can though they can be lucky or unlucky we dont know, 2- they will share what they learn from others, and they note whatever they want, since they know who they group with,they know which question answer is which, such as 1-A 20-D... – Oray Aug 11 '18 at 16:42
• (1) Are you allowed to bring 3 other students with you to the board at the same time or one at a time, but on up to 3 occasions? (2) Just to be sure, A and B come to the board for an exchange, and C, who is not involved in this exchange, will lose a point - right? – Heimdall Aug 18 '18 at 10:05
• @Heimdall they are together and share everything they have – Oray Aug 18 '18 at 10:16
• @Oray So that answers (1) up to 4 people gather at the board at the time, if I understand right. But (2) - is it just people that are board who lose a point or does everyone in the class lose a point? – Heimdall Aug 19 '18 at 9:23

What is the maximum average point they can get as a class in the worst case scenario?

I believe the solution for this question is:

107.875

This is with:

31 students scoring 32 points and 1 student scoring 31. Students share notes 20 times

For this strategy we must first realise that whenever a student receives notes, they are then capable of sharing those new notes as well as their starting note to other students. Additionally, we should realise that in order to gain the greatest benefit from any group of students sharing notes it is optimal to maximise the number of different notes between each pair of students in the group. With this in mind, we can perform the following:

Begin by splitting the students in to 10 groups of 3, leaving two students left over. For convenience, lets call these students Alice and Bob. Set Bob aside for now and have Alice sequentially share notes with each of the 10 groups. At the end of this process, each group of these groups will know 4, 7, 10, 13, ... , 31 answers with Alice also knowing 31 answers and Bob knowing 1 answer (the one that Alice doesn't know).

Next, remove one student from the first group (the group which knows 4 answers), replacing them with Bob and have Alice share notes with that group. This group now knows all 32 answers. Have Alice continue sharing notes with each other group until she reaches the group that knows 31 answers.

If Alice were to share notes with this group, three students would learn 1 correct answer where each correct answer is worth 4 points. So the class would only gain 12 points but it would cost the class 32 points meaning it is not worth it to share notes with this group. However, if we replace one of these students with the student Bob replaced (who still only knows 4 answers), sharing notes will net the class 120 points making it worthwhile to share notes.

At this point, sharing notes with the last remaining student is not worthwhile, so we should not do so.

What is the highest point they can get if every student wants to know all answers in the worst case scenario?

I believe the solution for this question is:

3,424 (giving a class average of 107)

The strategy here is the same as above but we share notes with the final student even though it nets a negative amount of points to the class.

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