Five students each answered five questions on an quiz consisting of two multiple-choice questions (A, B or C) and three True-False questions. They answered the questions as follows:

Student Q1 Q2 Q3 Q4 Q5 
Alex    A  A  T  T  T 
Bert    B  B  T  F  T 
Carl    A  B  T  T  F 
Dave    B  C  T  T  F 
Eddy    C  A  F  T  T 

No two students got the same number of correct answers. Who got the most correct answers?

Dave got the most correct answers(4), then
Bert(3)
Carl(2)
Alex(1)
Eddy(0)
Correct answers:
Q1= B
Q2=C
Q3=T
Q4=F
Q5=f

Solution:

I started with thinking about possible rankings of 5 students (who is the best and so on). So I made table with all possible differences of positions of different students. For examle if we consider Alex and Bert we will find out that the difference between them could be 1,2 or 3 points.

I was thinking about results in two parts. First I thout about True or False part. If answers of two students are different on one position then the difference in points from that part (betweene those two students) is exactly one point. If answers from T/F part are different on two positions then these two students could earn the same amout of points, or the difference between them could be 2 points.

T/F table (possible differences in points between students)
Alex Bert Carl Dave Eddy
Alex - (1) (1) (1) (1)
Bert - (0,2) (0,2) (0,2)
Carl - (0) (0,2)
Dave - (0,2)

Then I made similar table for first part of the test. You have to watch out becouse even when two students had completely different answers to both questions they could earn the same amout of points from that part (they all could be wrong).

Possible differences in points between atudents from ABC part
Alex Bert Carl Dave Eddy
Alex - (012) (01) (012) (01)
Bert - (01) (01) (012)
Carl - (012) (012)
Dave (012)

To compile these to tables we can't forget that to have all possible differences in points between students we should add or substract all combinations of numbers from first and second table. Of course I will not write 0 as a difference becouse we know that no two students got the same number of correct answers.

All possible differences in points.
Alex Bert Carl Dave Eddy
Alex - (123) (12) (123) (12)
Bert - (123) (123) (1234)
Carl - (12) (1234)
Dave - (1234)

From above table we can see that the biggest difference in points between any of two students can be 4 so there are only two possibilities of making ranking. 1. The best student made one mistake and the worst earned 0 points. 2. The best student made no mistake and the worst one made 4 mistakes. All other options are impossible as there had to be 5 points difference between the best and the worst student.

Eddy is for sure person who is the worst or the best becouse he is the only person with whom any of other students can have 4 points difference.

Eddy also had to make at least one mistake in ABC part becouse if he didn't make any mistake in that part then correct answers would be Q1=C Q2=A and then Carl and Dave would have two mistakes in part one (so the same amount of points from that part) and they have the same amount of points from second T/F part (becouse they have written the same answers) so they would have the same amount of correct answers which is not possible.

Let's consider each possible option.
I. Eddy is the worst with one good answer to question Q1.
Correct set of answers is
Q1 = C
Q2 = anything but A
Q3 = T
Q4 = F
Q5 = F
Then the students could earn:
Alex = 1 point
Bert = 2,3 points
Carl = 2,3 points
Dave = 2,3 points
That's impossible option because there would be two students with the same amount of points.

II. Eddy is the worst with one good answer to question Q2
Correct set of answers is
Q1 = anything but C
Q2 = A
Q3 = T
Q4 = F
Q5 = F
Then the students could earn:
Alex = 2,3 point
Bert = 2,3 points
Carl = 2,3 points
Dave = 2,3 points
That's impossible option because there would be two students with the same amount of points.

III. Eddy is the worst with one good answer to question Q3
Correct set of answers is
Q1 = anything but C
Q2 = anything but A
Q3 = F
Q4 = F
Q5 = F
Alex could earn = 0,1 point
That's impossible option because Eddy is the worst and has one correct answer so Alex would have the same number of correct answers.

IV. Eddy is the worst with one good answer to question Q4
Correct set of answers is
Q1 = anything but C
Q2 = anything but A
Q3 = T
Q4 = T
Q5 = F
Then the students could earn:
Alex = 2,3 point
Bert = 1,2,3 points
Carl = 3,4,5 points
Dave = 3,4,5 points
Someone has to have all correct answers.
If it is Dave who have all correct answers then
Alex = 2 points
Bert = 2 points
Impossible - no two students can have the same number of points.
If it is Carl who have all correct answers then
Alex = 3 points
Dave = 3 points
Impossible - no two students can have the same number of points.
So this whole option is impossible as well.

V. Eddy is the worst with one good answer to question Q4
Correct set of answers is
Q1 = anything but C
Q2 = anything but A
Q3 = T
Q4 = F
Q5 = T
The only person who has T/F part made all correct is Bert so he is the only person who wrote all correct answers (there must be someone with all corect answers, becouse Eddy is the worst and have 4 mistakes). But when we assume that Bert is the best then we can easily find out that Alex, Carl and Dave gained only 2 points what is impossible becouse no two students can have the same number of correct answers.

VI and VII. Eddy is the best one with on mistake (he had to make at least on mistake in part ABC, becouse otherwise Carl and Dave would have the same number of correct answers - which is impossible - see explanation above).
If Eddy made one mistake in ABC part then he made no mistakes in T/F part. You can see that there is no student who made 3 mistakes in part T/F so everyone has at least 1 point what is impossible because someone had to make mistake in every question (because Eddy is the best and has only 4 good answers).
That option is impossible then.

Last option is the only one possible.
VIII. Eddy is the worst with no correct answers
Correct set of answers is
Q1 = anything but C
Q2 = anything but A
Q3 = T
Q4 = F
Q5 = F
The only person who could make 4 mistakes is Alex, so
Correct set of answers is
Q1 = anything but C, anything but A
Q2 = anything but A
Q3 = T
Q4 = F
Q5 = F
The only person who could make 3 mistakes is then Carl, so
Correct set of answers is
Q1 = anything but C, anything but A
Q2 = anything but A, anything but B
Q3 = T
Q4 = F
Q5 = F
Then Bert made 2 mistakes and Dave made one mistake.
So Dave got the most correct answers.

Posted by Rafal on 2004-09-04 12:01:11