# Weird exam with odd number of questions

There are odd number of questions in an exam:

• Every question is answered correctly by at least one of the students in the class.
• Every student answers correctly an even number of questions.
• The number of questions correctly answered by any two students is even.

What is the minimum number of questions with the given condition above?

• the number of questions answered or the number assigned to the question asked?? Ie A answers questions 2 and 3- is the value you're looking for 2 or 5? Aug 8, 2017 at 2:09
• Third point do not add any significant information as it can be written as "The number of questions correctly answered by any number of students is even." Aug 8, 2017 at 7:15
• @NikhilBhavar without third info, the answer would be 3.
– Oray
Aug 8, 2017 at 7:25
• Second point says, every student answers even number of questions. And we know sum of odd/even number of even numbers is even. Which makes the third point look redundant. Aug 8, 2017 at 7:39
• So the third point is saying that that the number of distinct questions answered among two students is always even? I took it the same way as Nikhil Bhavar, so I think it's a bit unclear or ambiguous currently. Aug 17, 2017 at 8:18

Minimum questions is:

7

Because

Student 1 answers 1, 2, 3, and 4 correctly
Student 2 answers 3, 4, 5, and 6 correctly
Student 3 answers 1, 3, 5, and 7 correctly

✔ Number of questions is odd.
✔ Each question is answered correctly at least once.
✔ Every student answers four correctly; 4 is an even number.
✔ Any two students answer an even number of questions -
1+2 answer 1, 2, 3, 4, 5, and 6 - six questions is even.
1+3 answer 1, 2, 3, 4, 5, and 7 - six questions is even.
2+3 answer 1, 3, 4, 5, 6, and 7 - six questions is even.

It can't be less, because -

The number of questions must be odd.
• It can't be 1, as then the only even number of questions students could answer would be zero, and if all answer zero, not all questions get answered.
• It can't be 3, as then there are only three combinations of an even number of correct answers available to any student (we exclude "0"): 1+2, 1+3, and 2+3. At least two of these are needed to cover all 3 questions, and there is no way to pick two without there being a pair of students whose combined answers is 3, and thus not even.
• It can't be 5; each student could have two or four answers, but any combination of answerers that includes a student with four correct answers means that student plus a student with the fifth answer would make those two students a pair whose combined answers is 5, and thus not even - so each student can answer at most 2. You then run into a parallel scenario as for 3 questions, where it is always possible to find at least one pair of two answers out of any set of pairs that covers all of 1 to 5 that overlaps on one value, resulting in a pair of students whose combined answers is 3, and thus not even.

• Technically, this answer could be considered incorrect as the statement "The number of students is unknown." is not obviously true. But if you added an alternative solution with 6 students (student n and student n+3 could both have the same answers for example) and seven questions, you would be right. Aug 7, 2017 at 23:50
• Oray has apparently removed that statement now. :)
– Rubio
Aug 8, 2017 at 1:27