Question

A test consists of 30 true or false questions. After the test (answering all 30 questions), Mr Q gets his score: the number of correct answers. Mr Q doesn’t know any answer, but is allowed to take the same test several times. Can Mr Q work out a strategy that guarantees that he can figure out all the answers after the 29th attempt? after the 24th attempt?

Answer 1

about the first part shouldnt it be 30 attempts

Answer 2

hmm how about if i answer first question in first attempt then in second attempt i answer two questions (from the first attempt i already know what the correct answer for the first question is) and i do that until the 24th question or 24th attempt

Answer 3

with each attempt i ensure a correct answer for a single question

oh sorry i just realized it's 30 question test not 24

Answer 4

Smart! @Ishaan :D

Answer 5

Your strategy will work after 30 attempts, right?

Answer 6

yeah after 30 attempts
lol I need to rethink the strategy now

Answer 7

how about if I answer 2 question in my first attempt instead of one now lets say i have answered True for both of them
answered actual answer
1. t f f t t
2. t f t f t

now if obtain a zero or a full score from my first attempt, i get to know the actual answers

but what if i get only one right
hmm then i need to do something in second attempt

now if i get only one right in my first attempt i get to know that one of them is right or true and the other one is false, so in my next attempt

1. t t f t f
2 .f f t f t
3. t t f f t

now i am gonna get score of zero, 1 or 3, and any of the answer must confirm the answer of all three

hence, i have reduced it to 29th attempt

Answer 8

"......any of the score must confirm the answer of all three."

Answer 9

I think I might have figured it out for 24 attempts. I will try type it out now, not 100% sure of the answer yet, but I think it works.

Answer 10

now we gotta think if it's possible for 24 attempts

let's say i answer 4 questions in my second attempt
i am assuming that i got a score of 1 from my first attempt, i.e one of them is true and other is false, and i answered true for both of them in my first attempt


1. t t t f f t t f f
2. f f f t t f f t t
3. t f t f t t t f f
4. t t t f f f t f t

now i am gonna get either 0 or 2 for first two questions and either of the score is gonna confirm the answer for them....

now for 3 and 4 either i am gonna get 2 or 0 or 1
if it's 2 or 0 then i get to know the answer for 3 and 4th as well but if it's 1 then umm
for the next i.e 3rd attempt i am gonna do the same thing i did for the second attempt.

i am confirming answers for two questions with each attempt excluding the first attempt
confirmed answers
1. 0
2. 2
3. 2
4.... ....

interesting after 16 attempts i will be able to get the answers hmm i think i did something wrong here or maybe i assumed some stuff i shouldn't have. please do correct me

Answer 11

oh yeah i ignored some cases possible, please ignore me the solution i posted for fewer attempts is wrong

Answer 12

i am gonna go check other sections and questions :-)

good luck slaaibak

Answer 13

Okay, since we may only use 24 attempts, we must realize that we have 4 attempts for 5 questions.

Since 24/30 = 4/5

Okay, now that allows us to have 4 attempts for 5 questions, so if we can find a way to solve this, we can repeat is 6 times, hence answering 6 * 5 questions with 6 * 4 attempts.

Now, lets look at 5 questions.

First attempt:

Answer true for all the questions:

NOTE: EVERYTHING IS BASED ON THE FIRST ATTEMPT OF ANSWERING EVERYTHING TRUE!

Now there are 6 cases:

CASE1; 5 correct, 0 incorrect
CASE2; 4 correct ,1 incorrect
CASE3; 3 correct, 2 incorrect
CASE4; 2 correct, 3 incorrect
CASE5; 1 correct, 4 incorrect
CASE6: 0 correct, 5 incorrect

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CASE1

You get all five correct. Now you only need one guess, so it's solved for this case.

CASE2
----------------------------------------------------------
You get 4 correct.

Now what you do is:

Take the first three questions (1,2,3) and answer true to all of them.

Once again, there are two cases:

First case: All of them are true. Now you know the incorrect one is in the remaining 2 questions (question 4 and 5)

Now you just answer one of them true. If it's false, you know the other one is true. If it's true, you know the other one is true.
So you used three attempts until this, so it still works.

Second case: One of the three initial is false.

Now take the first two (1,2) and write both as true

Again 2 cases:

Case1: You get one of them wrong

Now take the first one and answer as true. If you are correct, you know the second question is false. If you are wrong, you know the first question is false and the second one is true.

You used 4 attempts, so it works.


Case2: You get both correct

Now you know that q3 is wrong, therefore false.

You used 3 attempts, so it works.


----------------------------------------------------------

CASE3

You get 3 correct and 2 incorrect. Now it gets interesting.


Second attempt (remember the first attempt was answering true for all)


Answer true for 1,2,3

Three cases:

Case1: All three are true. Therefore 4 and 5 is false. It's solved

Case2: Only one of the three is correct. Therefore you know 4 and 5 is correct. Now you have 2 attempts to figure out which one of 1,2,3 is correct.

Take 1 and 2 and answer both true. If both are incorrect, three is true and 1,2 is false. If one of them are incorrect, 3 is false. Now you need to figure out whether 1 or 2 is true. answer true to 1. If you are correct, 1 is true and 2 is false. If you are incorrect, 1 is false and 2 is true.

Now comes the third case -> The most difficult part in this solution.

case3:

2 of the three in 1,2,3 are true. This means 1 of 4,5 is also true.

Now for the solution:

Attempt number three: Take 3 and 4 and answer both true:

THREE CASES:

CASE 1: You get both correct. Now this means 5 is false. This also means one of 1,2
is true. Use the last attempt to say 1 is true. If you're correct, 2 and 5 is false, and 1,3,4 is true. If you're incorrect, 2,3,4 is true. and 1,5 is false.

CASE 2: You get both incorrect. This means 1,2,5 is true and 3,4 is false.

CASE 3: You get one correct. Now, you don't know which one you got correct. NOW! Take 4 and 1 and answer both true. This is your fourth attempt.

IF you get both correct, you know 4 MUST be true, leaving 3 FALSE. THEREFORE 5 is false. 1,2 and 4 is true while 3 and 5 is false.
IF you get none correct, you know 4 and 1 is false and 2,3,5 is true.
IF you get 1 correct, you know 4 false (since there's only one false answer in the first three) This implies 2 and 4 is false, while 1,3,5 is true.

That's it for CASE3!

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CASE4 uses the EXACT same logic as CASE3, since 3 are false and 2 are true. I'm not going to strain my brain to type it out, since CASE3 proves CASE4.

-----------------------------------------------------------------
CASE5 uses the same logic as CASE2, since 4 are false and 1 is true.

So I'm not going to type it out again.


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CASE6

Although this is the same logic as CASE 1, I'll type it out since it's short xD

You get all 5 wrong.
Now you know all 5 are false.


4 attempts works for all the cases. Hence you only need 4 attempts for 5 questions, which implies you only need 24 attempts for 30 questions.

I hope I didn't overlook anything!

Answer 14

I do not think that it's totally okay to say that it's been reduced to 5 questions, because you can't see the results for only five questions at a time...

Answer 15

You can. Just don't fill in the other 25 questions. Therefore it will work.

Answer 16

But how would you know a priori which of your five cases pertains to a set of five questions?

Answer 17

Oh I see, nevermind

Answer 18

Looks good to me, good job

Answer 19

@slaaibak
ur answer rocks,

can we further say that,

if we have n sets of questions and we want to figure their answer in k ways,

this can be done if and only if,

k/n = p/q

where p and p are coprime,

and q-p=1