9
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  1. Rocky the dog is alive.
  2. Rocky the dog is dead.
  3. Exactly one of statements 6 and 9 is true.
  4. Exactly one of statements 2 and 6 is false.
  5. Statements 4, 5 and 10 are all false.
  6. Exactly one of statements 1 and 10 is false.
  7. Exactly 5 statements are true.
  8. Exactly one of statements 3 and 10 is false.
  9. Exactly one of statements 6 and 10 is true.
  10. Exactly one of statements 1 and 2 is false.
  11. Statements 1, 8 and 11 are all false.

Which of the above statements are definitely true? Is Rocky the dog alive or dead?

The puzzle is based on a similar puzzle here: http://blog.physicsworld.com/2013/10/07/physics-world-at-25-puzzle-2/

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5
  • $\begingroup$ Statement 5 is a paradox. (Not enough rep to comment). $\endgroup$ Nov 8, 2014 at 14:22
  • $\begingroup$ Nice puzzle. It would be much harder if I weren't able to just try something and see what the consequences are $\endgroup$
    – d'alar'cop
    Nov 8, 2014 at 14:40
  • 4
    $\begingroup$ @KunalGupta Statement 5 is not paradoxical, but simply false (it cannot truly say about itself that it is false). As a consequence, statements 4, 5 and 10 are not all false, thus either statement 4 is true or statement 10 is true. This is similar to what happens with the so-called Epimenides paradox: it is no real contradiction (pace Russell), but simply a false statement which informs us that some other Cretan is a truth-teller. $\endgroup$
    – J Marcos
    Nov 9, 2014 at 3:48
  • $\begingroup$ @JMarcos It's also possible that both 4 and 10 are true. 5 would still be false. $\endgroup$
    – Kevin
    Nov 10, 2014 at 6:49
  • $\begingroup$ @Kevin, yes but strictly logically speaking A or B includes the possibility of A and B. msdn.microsoft.com/en-us/library/f355wky8.aspx $\endgroup$
    – Kenshin
    Nov 10, 2014 at 7:01

5 Answers 5

5
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The dog is:

Alive

Rocky the dog is alive.                         TRUE                
Rocky the dog is dead.                          FALSE
Exactly one of statements 6 and 9 is true.      TRUE
Exactly one of statements 2 and 6 is false.     FALSE
Statements 4, 5 and 10 are all false.           FALSE
Exactly one of statements 1 and 10 is false.    FALSE
Exactly 5 statements are true.                  TRUE
Exactly one of statements 3 and 10 is false.    FALSE
Exactly one of statements 6 and 10 is true.     TRUE
Exactly one of statements 1 and 2 is false.     TRUE
Statements 1, 8 and 11 are all false.           FALSE
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5
  • 1
    $\begingroup$ this is a valid solution, but does statement 7 have to be true? $\endgroup$
    – Kenshin
    Nov 9, 2014 at 8:26
  • 1
    $\begingroup$ @Mew No, funnily let's say statement 7 is a logical formula $a$. Then we have $ a\rightarrow a $. So it's true if it's true and false if it's false. $\endgroup$
    – d'alar'cop
    Nov 9, 2014 at 8:33
  • 2
    $\begingroup$ @Mew * $a\leftrightarrow a$ $\endgroup$
    – d'alar'cop
    Nov 9, 2014 at 21:50
  • $\begingroup$ @Mew I think an answer that simply states a solution without any explanation is not a good answer. I think there are better one you can accept. $\endgroup$
    – miracle173
    Nov 11, 2014 at 16:44
  • 1
    $\begingroup$ @miracle173, I don't care, I picked this one for good reason. $\endgroup$
    – Kenshin
    Nov 12, 2014 at 6:09
6
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Going down the list in order:

This whole line of reasoning is predicated upon 10 being absolute truth, because no dog can be both dead and alive. If that is not the case and we have a quantum dog that can be in a superposition of states, none of this is valid.

1 is a simple statement.
2 is a simple statement.

3 says that only one of 6 or 9 is true. If 3 is false, then either both 6 and 9 are true, or both 6 and 9 are false. Since statement 10 is true, if 9 is true, then 6 must be false. Therefore, they cannot both be true. IF 9 is false, then 6 must be true, otherwise 9 would be true. Therefore, they cannot both be false. Therefore, statement 3 must be true. Therefore, either 6 is true, or 9 is true, but not both.

9 states that only one of 6 or 10 is true. 10 tells us something we already knew to be true; therefore, if 9 is true, 10 is true and 6 is false. (We can also derive this from 3 being true, which makes 6 false if 9 is true). If 6 is false, then either 1 and 10 are both true, or they are both false. 10 is true, therefore 1 must be true; therefore, if 6 is false, rocky is alive.

If 6 is the one that is true, either 1 is true or 10 is true. 10 must be true, as 1 and 2 are logically opposed. Therefore, if 6 is true, 1 is false, making 2 true. Therefore, if 6 is true, Rocky is Dead.

On to the next statement. 4 states that one of 2 and 6 is false. If 2 is false, Rocky is Alive. However, if 2 is false, 6 is true, and if 6 is true, as we have shown, 2 is true, and Rocky is Dead. Therefore, if 2 is false, 2 is true, which is a logical contradiction. Therefore, 2 is true, and 6 is false. Therefore, if 4 is true, Rocky is Dead.

If statement 4 is false, then either 2 and 6 are both true or 2 and 6 are both false. If 2 is false (therefore Rocky is Alive) and 6 is false (therefore making statement 1 true, since 10 cannot be false for them both to be false), then Rocky is alive.

To recap the state of Rocky:
if 1: alive; if !1: dead
if 2: dead; if !2: alive
3, therefore either dead or alive.
if 4: Dead; if !4: alive
if 6: dead; if !6: alive.

If Statement 5 is true, then statements 4, 5, and 10 are all false. Since statement 5 being true precludes statement 5 being false, then statement 5 is false. This lets us know that statements 4, 5, and 10 are not all false. Statement 5 is false, so both of statements 4 and 10 cannot be false. Therefore, one ore both of statements 4 and 10 is true. However, we know that statement 10 is true. Therefore, either statement 4 is false or statement 4 is true. This tells us nothing of interest about statement 4; we already knew it was either true or false.

Statement 7 claims that five statements are true. This is not useful at this time, so we set it aside for now.

Statement 8 claims that exactly one of 3 and 10 is false. We know that 10 is empirically true. Therefore, if 8 is true 3 is false. We know, however, that 3 must be true. Therefore, 8 is false. You can reason out here to show that both statements 3 and 10 are true, but we knew that already.

To recap the state of Rocky:
if 1: alive
if 2: dead
since 3, either Dead or alive
if 4: Dead; if !4: alive
if 6: dead
since !8, either dead or alive


Statement 11 claims that 1, 8, and 11 are all false. As with statement 5, 11 being true leads to a logical contradiction, so we know that 11 is false. Since we know that 11 is false and 8 is false, 1 must be true (otherwise 11 would be true). Therefore, 1 is true and rocky is ALIVE. Since he is alive, he cannot be dead, so 2 is false.

So far:
1: true
2: false
3: true
5: false
8: false
10: true
11: false

We deduced earlier that if statement 4 is true, then rocky is dead. Since rocky is not dead, statement 4 must be false. We deduced earlier that that means 2 and 6 must both be true or both be false; since 2 is true, 6 must be false.

So far:
1: true
2: false
3: true
4: false
5: false
6: false
8: false
10: true
11: false
Statement 9 claims that exactly one of 6 and 10 is true. This is obviously true.

1: true
2: false
3: true
4: false
5: false
6: false
8: false
9: true
10: true
11: false

We now have four true statements. Statement 7 states that exactly 5 statements are true. If 7 is true, then five statements are true, as 7 is the fifth. If 7 is false, then ther are only four true statements, making 7 false. So 7 is impossible to determine. I will mark this as "mu".

Final answer:

Rocky is Alive.
1: true
2: false
3: true
4: false
5: false
6: false
7: mu
8: false
9: true
10: true
11: false

As a side note, I am ridiculously proud that I managed to avoid confusing myself, but please let me know if I mixed something up in the middle!

Edited: I had mixed up statement 8 somehow. Expanded and fixed.

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4
  • $\begingroup$ you have mixed something up somewhere. Your final answer is also wrong (by mistype I assume) because you have both statement 1 and 2 being true, meaning he is alive and dead at the same time. $\endgroup$
    – Kenshin
    Nov 9, 2014 at 3:59
  • $\begingroup$ @Mew Typo fixed. I'll revisit the answer tomorrow, see if I can pick out what I did wrong $\endgroup$ Nov 9, 2014 at 4:01
  • 1
    $\begingroup$ Note that 7 is a good clue you've mixed something up: you presently have it listed as 'false' while it expresses the true statement that there are five true answers. With your current assignment of answers to the rest of the questions there's no way of making 7 correct. $\endgroup$ Nov 9, 2014 at 17:41
  • $\begingroup$ @Gilles It's Schrodinger's dog! $\endgroup$ Jun 23, 2015 at 22:22
3
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There are 2 solutions:

1. T, 2. F, 3. T, 4. F, 5. F, 6. F, 7. T, 8. F, 9. T, 10. T, 11. F
1. T, 2. F, 3. T, 4. F, 5. F, 6. F, 7. F, 8. F, 9. T, 10. T, 11. F
So, answers 1,3,9 and 10 are definitely true and Rocky is alive.

And here is the long story why

10. "Exactly one of statements 1 and 2 is false."
Rocky is alive or dead. 10 is true.

9. "Exactly one of statements 6 and 10 is true."
10 is true, so it becomes "Exactly zero of statements 6 is true." or "6 is false".
6 and 9 have opposite truth values.

3. "Exactly one of statements 6 and 9 is true."
3 is true because of the statement just above.

8. "Exactly one of statements 3 and 10 is false."
10 is true, 3 is true, so 8 is false.

11. "Statements 1, 8 and 11 are all false."
If it were true it would say it is false. => 11 is false
To be false, one of 1, 8, 11 must be true. 8 and 11 are false, therefore 1 must be true.
This proves Rocky is alive. 1 is true and 2 is false.

5. "Statements 4, 5 and 10 are all false."
10 is true, so 5 is false.

6. "Exactly one of statements 1 and 10 is false."
1 and 10 are true, so 6 is false. And 9 is the opposite of 6, 9 is true.

4. "Exactly one of statements 2 and 6 is false."
2 and 6 are false, so 4 is false.

7. Exactly 5 statements are true.
So far, we have:
1 true, 2 false, 3 true, 4 false, 5 false, 6 false, 7 ?, 8 false, 9 true, 10 true, 11 false.
Besides 7, four statements are true.
If 7 is true then we have five true statements and 7 evaluates to true.
If 7 is false then we have four true statements and 7 evaluates to false.
7 can either true or false.

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5
  • $\begingroup$ @Mew You are right, the comment was based on a wrong answer. I fixed that. $\endgroup$
    – Florian F
    Nov 9, 2014 at 7:59
  • 1
    $\begingroup$ the last 2 answers you have a wrong. Statement 1, 8 and 11 are all false, but this means statement 11 must be true, which is a contradiction. $\endgroup$
    – Kenshin
    Nov 9, 2014 at 8:08
  • $\begingroup$ @Mew you are right again. I thought I checked each statement. $\endgroup$
    – Florian F
    Nov 9, 2014 at 13:15
  • $\begingroup$ I reworked the answer. The solution have been given by others but the explanation is easier to follow I think. $\endgroup$
    – Florian F
    Nov 9, 2014 at 13:53
  • $\begingroup$ I see this is correct, good work +1. $\endgroup$
    – Kenshin
    Nov 9, 2014 at 13:54
1
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Statement 5 tells us that 4 and 10 are true. So, it confirms that the dog is dead or alive ( statement 1 and 2 ). If 4 is true ,we can find a conflict between 6 and 10. Because, if 10 is false, it is not possible that 1 and 2 are true at the same time, either false. So 10 must be true and 1 is false, so the dog is dead
The true statement are: 2, 3, 4, 6, 8, 10

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5
  • 1
    $\begingroup$ Are you sure that both 4 and 10 must be true? (I think 5 implies that at least one of them is true) $\endgroup$
    – d'alar'cop
    Nov 8, 2014 at 14:29
  • $\begingroup$ I understood that "all false" could be reverted in "all true", but I can be wrong! ^^ $\endgroup$
    – Emi987
    Nov 8, 2014 at 14:32
  • $\begingroup$ @Emi987, I think you have misinterpreted statement 5. $\endgroup$
    – Kenshin
    Nov 8, 2014 at 14:34
  • $\begingroup$ @Mew I see! That's fine, my bad! $\endgroup$
    – Emi987
    Nov 8, 2014 at 14:36
  • 3
    $\begingroup$ en.wikipedia.org/wiki/Universal_quantification#Negation $\endgroup$ Nov 8, 2014 at 15:04
-1
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My calculations say Statement 2 is true. Fingers crossed...

Reason for this answer... Statement 5 and 11 are themselves false so no considered... As per statement 3, 6 & 9 any 1 is true again statement 9 says 6 & 10 any 1 is true but statement 6 which has probability of being true twice says that 1 & 10 is false. So if I consider both to be false for a while statement 10 says 1 & 2 any 1 is false

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2
  • $\begingroup$ Reason for this answer... Statement 5 and 11 are themselves false so no considered... As per statement 3, 6 & 9 any 1 is true again statement 9 says 6 & 10 any 1 is true but statement 6 which has probability of being true twice says that 1 & 10 is false. So if I consider both to be false for a while statement 10 says 1 & 2 any 1 is false $\endgroup$ Nov 8, 2014 at 16:36
  • 1
    $\begingroup$ Why dont you edit your answer rather than writing this in a comment $\endgroup$
    – skv
    Nov 10, 2014 at 6:59

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