How do you start?

Logic Level 4

"This test" refers to the following set of three questions. In particular, "questions in this test" can count itself.

  1. How many questions in this test (including this) have a different answer from this question?
  2. How many questions in this test (including this) have the same answer as this question?
  3. What is the square of the answer to this question?

Concatenate the answers to the three questions in order. For example, if the answers are 0 , 1 , 2 0, 1, 2 for Questions 1, 2, 3 respectively, then submit 012 = 12 012 = 12 .

This problem is created by Nikolai Beluhov .


The answer is 210.

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Ivan Koswara
Mar 14, 2015

By looking on each question individually first, we get the following:

  • Question 1 has an answer in { 0 , 1 , 2 } \{0, 1, 2\} . There are three questions, so the answer must be one of 0 , 1 , 2 , 3 0, 1, 2, 3 , but Question 1 cannot count Question 1 itself as it clearly has the same answer as itself.
  • Question 2 has an answer in { 1 , 2 , 3 } \{1, 2, 3\} . Again, there are three questions, but Question 2 must count itself.
  • Question 3 clearly has an answer of either 0 0 or 1 1 .

Now, seeing that no other question has an answer of 3 3 , Question 2 cannot have an answer of 3 3 as well. If Question 2 has an answer of 2 2 , then the only other question that can give an answer of 2 2 is Question 1, but Question 1 states that Question 2 and 3 must have different answers from Question 1, contradiction. Thus Question 2 has an answer of 1 1 ; this forbids any other question for an answer of 1 1 , so Question 3 has an answer of 0 0 . Finally, Question 1 cannot have an answer of 0 0 as Question 2 has a different answer from it, so Question 1 has an answer of 2 2 .

This gives the unique solution 2 , 1 , 0 2, 1, 0 , giving the answer 210 \boxed{210} .

Why can't it be 110?

Earl Adams - 5 years, 9 months ago

Log in to reply

There are 2, not 1, questions having answer the same as Q2 (namely Q1 and Q2).

Ivan Koswara - 5 years, 9 months ago

Log in to reply

Hmm. I don't think you should count the question itself. Maybe the problem should be worded to make that more clear.

Colin Carmody - 5 years, 4 months ago

Log in to reply

@Colin Carmody "How many questions in this test" quite unambiguously implies all questions in the test, including itself.

Ivan Koswara - 5 years, 4 months ago

Oh I see, I was not counting the question itself

Earl Adams - 5 years, 8 months ago

Why can Question 3 only be either 0 0 or 1 1 ?

Patrick Engelmann - 6 years, 1 month ago

Log in to reply

If the answer to Question 3 is x x , then Question 3 states that the answer is x 2 x^2 . This means x = x 2 x = x^2 , which is a simple algebra equation with solutions x = 0 , 1 x = 0, 1 .

Ivan Koswara - 6 years, 1 month ago

Log in to reply

I see. thanks!

Patrick Engelmann - 6 years, 1 month ago

it just says what is the square of the answer to this question. where did they sat they are equal?

raghavendra mananadi - 4 years, 8 months ago

Log in to reply

@Raghavendra Mananadi The answer of Q3 is the square of the answer to Q3. Of course the answer to Q3 is equal to the answer to Q3. If the answer to Q3 is x x , then its square is x 2 x^2 ; Q3 then says that the answer to Q3 is x 2 x^2 . But the answer to Q3 is also x x , so x = x 2 x = x^2 .

Ivan Koswara - 4 years, 8 months ago

If 2,1,0 is the answer, then from q3: 0*0=0 which is fine, from q1, the answer to q2 and q3 are different than q1 which is the case as 1 and 0 are both different than 2. But if the answer to q2 is 1, then there is one either q1 or q3 with the same answer as q2. Which is not the case... When you write q1 cannot count q1 itself, it also means that q2 cannot count q2 itself! If this is not the case, than you should specify it in the rules somehow.

However, 1, 1, 0 works according to that logic.

Benjamin Katz-Crowther - 5 years, 10 months ago

Log in to reply

Q1 cannot count itself since Q1's answer is the same as Q1's answer. If Q1 has answer x x , then among the three questions in the test, Q1 itself has answer x x and thus is not different from x x , hence it's not counted among the questions having answer different from Q1 (that is, different from x x ).

On the other hand, the opposite logic works for Q2. If Q2 has answer y y , then among the three questions in the test, Q2 itself has answer y y and is thus counted among the questions having answer the same as Q2 (that is, the answer y y ).

It is confusing. Self-referential statements are confusing.

Ivan Koswara - 5 years, 10 months ago

101 also works!!

Shreyansh Singh Solanki - 5 years, 9 months ago

Log in to reply

Your answer to Question 2 would be wrong then: there is 1, not 0, question having the same answer to Question 2 (namely Question 2 itself).

Ivan Koswara - 5 years, 9 months ago

Log in to reply

GOT IT NOW ;)

Shreyansh Singh Solanki - 5 years, 9 months ago

Please add that Q2 counts itself in "Assumptions". If you are trying to solve it without that assumption, you get 201, 101 and 110 as right answers. Which I all tried :(

Sasha Chabanov - 5 years ago

Log in to reply

@Sasha Chabanov See my reply in another comment:

"How many questions in this test" quite unambiguously implies all questions in the test, including itself.

I stated that "this test" is the three questions given, so "how many questions in this test" means "how many questions among all these three", which means it counts itself.

Ivan Koswara - 5 years ago

You stated in your solution that "Question 1 cannot count Question 1 itself as it clearly has the same answer as itself." But for question 2, you don´t use the same logic, as you did count the answer on question 2. For that matter, i found 101 to be correct also.

Yan Coelho - 5 years, 9 months ago

Log in to reply

@Yan Coelho Exactly; Q1 cannot count itself, since Q1 always has the same answer as Q1. Q2 must count itself, since Q2 always has the same answer as Q2.

1,0,1 doesn't work; there is one, not zero, question having the same answer as Q2 (namely Q2 itself).

Ivan Koswara - 5 years, 9 months ago

Why not 211?

Mbah Abal - 5 years, 4 months ago

Log in to reply

Question 2 is thus wrong. There are two questions with answer 1 (namely Q2 and Q3).

Ivan Koswara - 5 years, 4 months ago

How does "Question 1 states that Question 2 and 3 must have different answers from Question 1". It never says they have to all be different

Jon Venne - 5 years, 11 months ago

Log in to reply

If Question 1 has an answer 2 2 , then the only way to get this is if Questions 2-3 have different answers from Question 1 (otherwise there are less question with different answers).

Ivan Koswara - 5 years, 11 months ago

Why can't 201 be an answer?

Piyush Ravi - 5 years, 10 months ago

Log in to reply

There is 1 question having the same answer to Q2 (namely Q2 itself), not 0.

Ivan Koswara - 5 years, 10 months ago

Why isn't 219 correct?

Steven Feeser - 5 years, 7 months ago

Log in to reply

Question 3 is then wrong; the square of Question 3's answer is 81, not 9.

Ivan Koswara - 5 years, 7 months ago

Log in to reply

oh..thanks..:D

Steven Feeser - 5 years, 7 months ago

Why can it not be 101

Dnumyar Bonoan - 5 years, 6 months ago

Log in to reply

Your answer to Question 2 is wrong (there is one, not zero, question having the same answer as Question 2).

Ivan Koswara - 5 years, 6 months ago

Rubbish. Each question has one answer. But question number 2 asks how many questions have the same answer, which is none !! Simple logic tells you that if two or more questions have the same answer, they would be the same question !! The correct answer must be 201 = 21 !!

DarkMind S. - 4 years, 8 months ago

Log in to reply

There is one question having the same answer as Question 2, namely Question 2 itself. It doesn't say "how many other questions have the same answer"; it says "how many questions in this test have the same answer".

Ivan Koswara - 4 years, 8 months ago

Log in to reply

oh c'mooooon !!. It dosnt have to say it

DarkMind S. - 4 years, 8 months ago

" if two or more questions have the same answer, they would be the same question" - what?

So ... Question 1: How many arms do you have? Question 2: What's the square root of 4?

Would you consider those to be the same question??

Ban An - 4 years, 8 months ago

Question 2 is ambiguous. It might very well be read as "how many OTHER questions have the same answer as this question". The problem lies in the interpretation of "having the same answer".

If I'm in a room, and I say "how many people in this room have the same name as me", I am obviously not counting myself.

If you read question 2 like that, the answer is either 201, 110, 101, or 211. :)

Ban An - 4 years, 8 months ago

Log in to reply

I intentionally state "how many questions in this test have the same answer", and I've also said that "this test" refers to the three questions, not the two other questions.

Ivan Koswara - 4 years, 8 months ago

Log in to reply

Hey, we all make mistakes. Just admit that you blew it on this problem, and lets move on. Whats done is done

DarkMind S. - 4 years, 8 months ago

Log in to reply

@DarkMind S. No, I don't admit that I blow the problem, because I can likewise say that the people answering Q2 to read "other questions" to fail at reading. (I intentionally phrase the problem to make it unambiguous as possible, with "this test refers to the following three questions".) Nevertheless, I clarified it in the problem that all questions can count themselves. (And now I'll expect people thinking that Q1 counts itself.)

Ivan Koswara - 4 years, 8 months ago

why q3 is either 0 or 1

raghavendra mananadi - 4 years, 8 months ago

Hi, What about 121?

Jief Gre - 4 years, 7 months ago

Log in to reply

Q2's answer is wrong (only one question has the same answer as Q2).

Ivan Koswara - 4 years, 7 months ago

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...