Quandaries and Queries
Harry, Neville, Ron, Hermione and Malfoy each answered five questions on O.W.L. exam consisting of two multiple-choice questions (A, B or C) and three True-False questions. They answered the questions as follows:
If no two students got the same number of correct answers, who got the most correct answers? And what is the correct answer for each question?
If you look at Ha, N, R, He & M in pairs you'll notice that each shares the same number of answers with at least 2 others and as such no one could have all 5 answers correct if they each answered a different number correct. Thus they had 0,1,2,3 & 4 correct.
Who could have got 0 correct? Looking at the number of answers shared we see we cannot have Ha = 0 and, if N, R or He = 0 then M = 4. If M = 4 and M got the T, F's right then no one could have 0 correct. If M = 4 and the multiple choices were correct then R = He, impossible. Thus M = 0.
The answers therefore are A or B, B or C and T, F F. I think it's just as well now to try the 4 cases and they lead to B, C, T, F, F and Ha = 1, N = 3, R = 2, He = 4 and M = 0.
Step 1: Check that nobody could have gotten all answers correct. That means the possible scores must be 0 through 4.
Step 2: The person with 0 correct answers can have at most one matching answer with the person who got four correct answers. All pairs of answer lists having one match involve Malfoy, so he MUST have gotten them all wrong.
Step 3: GUESS that Hermione is the student with four correct answers. The one place she agrees with Malfoy's wrong answer is in the 4th answer, which would therefore have to be F. The correct answer scheme under this guess would be BCTFF.
Step 4: Confirm that this is correct. If not, return to step 3. (It is correct, but one must follow through all possibilities in step 3 to see if the solution is unique. I haven't.)