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Five Students

Puzzle:

Five students - Akash, Chintan, Jignesh, Mukund and Venky - appeared for an exam. There were total five questions - two multiple choice (a, b or c) and three true/false questions. They answered five questions each and answered as follow.


I

II

III

IV

V

Chintan

c

b

True

True

False

Akash

c

c

True

True

True

Jignesh

a

c

False

True

True

Mukund

b

a

True

True

False

Venky

b

b

True

False

Tru

Also, no two students got the same number of correct answers. Can you tell which are the correct answers? What are their individual score?

For Solution SCROLL DOWN...

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Solution:

The correct answers are b, a, True, False and False. Also, the scores are Jignesh (0), Akash (1), Chintan (2), Venky (3) and Mukund (4).

As no two students got the same number of correct answers, the total number of correct answers must be either 15 (1+2+3+4+5) or 10 (0+1+2+3+4).

Let's find out the maximum number of correct answers possible from the answers given by them.
For Question I = 2 (b or c)
For Question II = 2 (b or c)
For Question III = 4 (True)
For Question IV = 4 (True)
For Question V = 3 (True)

Thus, the maximum number of correct answers possible are 15 (2+2+4+4+3) which means that Akash would have given all correct answers as only he answered True for questions III, IV and V. But then Chintan and Jignesh would have exactly 3 correct answers. And also, Mukund and Venky would have 2 correct answers. So no one got all five correct. One can also arrive at this conclusion by trial-and-error, but that would be bit lengthy.

Now, it is clear that total number of correct answers are 10 (0+1+2+3+4). Questions III and IV both can not be False. If so, total number of correct answers would not be 10. So the student who got all wrong can not be Chintan, Akash and Mukund.

If Venky got all wrong, then Chintan, Jignesh and Mukund each would have atleast 2 correct answers. It means that Akash would have to be the student with only one correct answer and the correct answers for questions I and II would be a and a respectively. But then the total number of correct answers would be 1 (a) + 1 (a) + 1 (False) + 4 (True) + 2 (Flase) = 9.

Thus, Jignesh is the student with all wrong answers. The correct answers are b, a, True, False and False. Also, the scores are Jignesh (0), Akash (1), Chintan (2), Venky (3) and Mukund (4).


173 comments:

pg... said...

1) b
2) a
3) true
4) false
5) false.
mukund--->4
venky --->3
chintan-->2
akash---->1
jignesh-->0

pg... said...

Sol
They got different scores
So their scores might be (1,2,3,4,5) or (0,1,2,3,4)
case 1) Assume that Chintan has answered all the 5 questions correctly
then aakash and mukund will get same score 3
So Chintan didnt get score 5

case 2)if Aakash gets score 5 Chintan and Jignesh will get same score 3
so on ........ (1,2,3,4,5) is not the answer.

for (0,1,2,3,4)
case 1)
Assume that answers are c,b,tr,tr,tr then chintan and aakash will get same score 4
so this is not the answer

case2) Assume answers are c,b,t,f,f
then aakash and mukund will get the same score 2. So this is not the answer
so on ......
Chintan didnt get the score 4
........
if mukund gets the score 4 and the answers are b,a,t,f,f, all members will get different scores

Sudip said...

Let's assume someone answered all the Qs correctly.
The matrix below shows the score-chart ().
------------------C--A---J---M--V
C - c b T T F - 5 - 3 - 1 - 3 - 2
A - c c T T T - 3 - 5 - 3 - 2 - 2
J - a c F T T - 1 - 3 - 5 - 1 - 1
M - b a T T F - 3 - 2 - 1 - 5 - 2
V - b b T F T - 2 - 2 - 1 - 2 - 5
in each of the columns (or rows), at least one score appears more than once, so, none got all the answers right (i.e., nobody scored 5)
So someone must have got all the answers wrong (i.e., someone scored 0)
The answer combinations of the T-F Qs that have occurred are - FTT, TFT, TTF, TTT, one of which are all wrong answers, and the compliment of which r all correct.
so the possible correct cobinations are - TFF, FTF, FFT, FFF
if it was FFF, marks scored by the students for these 3 T-F Qs are
C - 1
A - 0
J - 1
M - 1
V - 1, but the highest scorer (who scores 4) has to score at least 2 in the last 3 Qs. So this is not a possible correct combination.

if it was TFF, marks scored by the students for these 3 T-F Qs are
C - 2
A - 1
J - 0
M - 2
V - 2
then, either cb(C highest scorer), ba(M highest scorer) or bb(V highest scorer) must be the possible correct answers for the first 2 Qs.
cb combination produces - 42023 (invalid)
ba combination produces - 21043 (which satisfies the criteria, and hence the answer)
So, the correct answers are baTFF

Suren said...

Couple answers posted here including the "official" one assume that if "no two students got the same number of correct answers." means the scores must be 1,2,3,4,5 or 0,1,2,3,4. But there are possibilities of havind combinations like 0,2,3,4,5; 0,1,3,4,5; etc. The fact that scores are 0,1,2,3,4 can be derived from the fact that no one got all answers right (which can be seen from the matrix posted above).

Suren said...

A little longer way, but to me it's more mathematically precise and explains a clear algorythm without guessing:

Step 1: As Sudip shows no one answered all questions correctly, so scores are 0,1,2,3,4.
Step 2. Let's define a function A(B) that returns the score for answering question A with answer B, A=I|II|III|IV|V and B=a|b|c|t|f. For example I(c) means the score for answering 'c' on the first question. Note that the function returns either 1 for correct answer or 0 for incorrect one.

Then the matrix can be rewritten as:
I(c)+II(b)+III(t)+IV(t)+V(t) = Sc (Chintan Score)
I(c)+II(c)+III(t)+IV(t)+V(t) = Sa
I(a)+II(c)+III(f)+IV(t)+V(t) = Sj
I(b)+II(a)+III(t)+IV(t)+V(f) = Sm
I(b)+II(b)+III(t)+IV(f)+V(t) = Sv

Sc!=Sa!=Sj!=Sm!=Sv => Sx-Sy!=0 for any x and y.

Sc-Sa=II(b)-II(c)+V(f)-V(t)!=0
Note that if II(b)=1 then II(c)=0 because only one of the answers for question II can be correct, similarly if V(f)=1 then V(t)=0. Considering these facts and the equasion the followinf can be derived:

if II(b)=1 then V(f)=1
if V(f)=1 then II(c)=0
if II(c)=1 then V(t)=1
if V(t)=1 then II(b)=0

Similarly from Sa-Sj we derive:

if I(c)=1 then III(t)=1
if III(t)=1 then I(a)=0
if I(a)=1 then III(f)=1
if III(f)=1 then I(c)=0

From Sc-Sm we derive:

if I(c)=1 then II(a)=0
if II(a)=1 then I(c)=0
if I(b)=1 then II(b)=0
if II(b)=1 then I(b)=0

From Sc-Sv we derive:

if IV(t)=V(t) (which means if answer to IV is the same as the answer to V) then I(c)!=I(b) => I(a)=0

From Sm-Sv we derive:

if IV(t)=V(t) then II(a)!=II(b) => II(c)=0

Step 3.
Let's inverse the original matrix. We know that one of the rows in the inversed matrix will be the correct answer (the one that is inversion of the answer that got 0 score). Using the rules derived in the Step 2 we ca eliminate some of the options. After that elimination the inverse matrix looks like:

C - a|b a|c f f t
A - b a|b f f f
J1- b a t f f
J2- c b t f f
M - a c f f t
V - a a f t f

Note that I split answers for J because in
J - b|c a|b t f f
b a t f f
b b t f f [not valid]
c a t f f [not valid]
c b t f f
The two options are not valid because of the rules derived from Sc-Sm.

Inverse matrix gives us 10 options:

C1 - a a f f t
C2 - a c f f t
C3 - b a f f t
C4 - b c f f t
A1 - b a f f f
A2 - b b f f f
J1 - b a t f f
J2 - c b t f f
M - a c f f t
V - a a f t f

Note that C2=M and therefore none of them can be the correct answer (otherwise both Chitan and Mukund would have 0 score). We narrowed down the options of the correct answer to 8. Good enough to just check which one of them is correct.
If C1 were correct answer Sa=Sm=1
If C3 were correct answer Sj=Sm=2
If C4 were correct answer Sj=Sv=3
If A1 were correct answer Sc=Sj=1
If A2 were correct answer Sc=Sm=2
J1 is correct answer Sc=2,Sa=1,Sj=0,Sm=4,Sv=3

Let's make sure it's the only one:
If J2 were correct answer Sa=Sm=2
If V were correct answer Sj=Sm=3

Therefore J1 (b a t f f) is the only correct answer.

Ufff. I'm gone to sleep ;)

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David Jackman said...

Wow, this one was tough! Let me give my answers and I'll explain my reasoning in as close to plain english as possible!

First, I got the correct answers as

Q1: B
Q2: A
Q3: True
Q4: False
Q5: False

and the scores of each student are:

C: 2
A: 1
J: 0
M: 4
V: 3.

Okay, now for my reasoning.

The rule here is that no two students can have the same number of correct answers. So we have 6 possible numbers of correct answers (0 correct answers, 1 correct answer, 2, 3, 4, or 5 correct answers).

Since no student got completely opposite answers as another student, either one student got all the answers right or one student got all the answers wrong, but not both.

So from this reasoning I asked if a student could have gotten 5 correct answers and another had 4 correct answers? The answer is no, because there are no two students that have 4 similar answers. Therefore, there had to be one student with a score of 0, one with a score of 1, one with 2, one with 3, and one with 4.

Now I wanted to find the extremes (because it was the only thing I think I could have found at this point). So I reasoned that the student with 4 correct answers and the student with 0 correct answers will have 1 similar answer (they could have had 0, but again, there were not 2 students with completely opposite answers). These pairs could have been

C & J
M & J or
V & J.

Each of the above pairs has just 1 similar answer between them. Now since the student who got 4 questions correct and the student who got 0 questions correct will have 1 similar answer, it stands to reason that J got either 0 questions correct or 4!

Now, it seemed unlikely that J could have gotten 4 correct answers and have 3 other students with 1 similar answer, so I made an educated guess that J had 0 correct answers. (If the logic didn't support my guess, I would have gone back and tried something else, but it turns out I was working down the right path.)

So with J getting 0 correct answers, the Correct Answers looked like

Correct Answers: Q1(B/C) Q2(A/B) Q3(T) Q4(F) Q5(F)

Now I looked at C, M, and V, given that they each had 1 similar answer to J:

C had 2-4 correct answers (depending on questions 1 & 2)
M had 2-4 correct answers
V had 2-4 correct answers.

Well at this point I decided to compare my correct answers to A.

A had 1-2 correct answers.

Now, working under the assumption that J had 0 correct answers and one student had to have had 1 correct answer, it must be A that had 1 correct answer because neither C, M, nor V can have just one correct answer!

So A has one correct answer (question 3). Therefore, A's answers on questions 1(C) & 2(C) are wrong. So my correct answers now become:

Correct Answers: Q1(B) Q2(A/B) Q3(T) Q4(F) Q5(F)

Now from here the only thing to do is see whether the answer to question 2 is A or B based on the consequences. If answer 2 is B, then both C and M have 3 correct answers, which conflicts with the rule laid out in the puzzle.

However, if answer 2 is A, then C has 2 correct answers, M has 4 correct answers and V has 3 correct answers! And we guessed J has 0 correct answers and A has 1 correct answer! It fits perfectly! Therefore the result must be correct!

Good puzzle!

Thinking said...

Ha
I should share this with some people who love puzzles