Another six-statements logic puzzle

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Cradarc
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Another six-statements logic puzzle

Postby Cradarc » Sat May 02, 2015 1:57 am UTC

A: F is true.
B: A is false.
C: B is false.
D: C is true.
E: A is true if D is true.
F: A is true if E is true.

Is it possible to determine which statements are true? If so, identify them.

Edit: Just to be clear, the answer should seem pretty straightforward, but then consider the following rephrased version:
A: F is true.
B: A is false.
C: B is false.
D: C is true.
E: A and D are both true.
F: A and E are both true.

Is the solution to this version different? Should the solution to this version be different?
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Wildcard
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Re: Another six-statements logic puzzle

Postby Wildcard » Sat May 02, 2015 3:45 am UTC

Spoiler:
E is true. The rest cannot be determined.

There are two possibilities, either
A, C, D, E, F are true and B is false, or
A, C, D, F are false and B, E are true. But either way E is true.

In your second version, the answers are: yes, the solution is different, and yes, they should be different. This is because "implies" (aka if/then) is a different logical relation than "and." The first version uses "if" and the second version uses "and."

For the second version, either
A, C, D, E, F are true and B is false, or
A, C, D, E, F are false and B is true.

So for the second version, it is not possible to determine which statements are true.
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Kingreaper
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Re: Another six-statements logic puzzle

Postby Kingreaper » Sun May 03, 2015 4:01 am UTC

For the original puzzle:
Spoiler:
E is true, because if D is true then C is true, B is false and A is true.

If D were true, then only B would be false (AbCDEF). If D were false then only B and E would be true (aBcdEf).

Both of these are consistent, until you look at F again; F says that if E is true then A is true.

But there is a possible solution (aBcdEf) wherein E is true and A isn't. This proves that F is false, and means that aBcdEf is the only possible solution.


For the modified version:

Spoiler:
The solution certainly could be different, as the problem is meaningfully different; for starters E is no longer trivially true.

AbCDEF is a valid solution to this problem, as is aBcdef. There is nothing to distinguish between the two; but neither are the same as the solution to the first problem, so yes the solution is different.

Cradarc
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Re: Another six-statements logic puzzle

Postby Cradarc » Sun May 03, 2015 5:46 am UTC

Kingreaper,
Spoiler:
Kingreaper wrote:But there is a possible solution (aBcdEf) wherein E is true and A isn't. This proves that F is false, and means that aBcdEf is the only possible solution.

Notice that f is false in that solution, so aren't you basically saying "f must be false because there is a consistent solution in where it is false"?

Suppose you assign truth values to every statement and feed it into a computer. The computer will not search for all the solutions, but merely check to see if your submitted answer is a solution. If you feed it AbCDEF, what will be the response?
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Re: Another six-statements logic puzzle

Postby Ralp » Thu May 07, 2015 3:26 pm UTC

Wildcard wrote:
Spoiler:
E is true. The rest cannot be determined.

Given this property, I think this puzzle fleshes out extremely well for a practical encounter in a roleplaying game, and I intend to steal it to that end. :twisted: An adventuring party exploring a crypt comes across a chamber with six talking skulls. Every skull knows the layout of the crypt and every skull either always tells the truth or always lies, so the adventurers have to choose one to bring along as a guide. Even if they know the chosen skull always lies they can just do the opposite, but grabbing an unknown skull could lead them to sudden death. "Skull F always tells the truth!" explains Skull A, and so on.

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SPACKlick
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Re: Another six-statements logic puzzle

Postby SPACKlick » Thu May 07, 2015 3:40 pm UTC

Ok original
Spoiler:
Two solutions
¬A, B ¬C, ¬D, E, ¬F
A, ¬B, C, D, E, F
In all cases E is true


Modified
Spoiler:
two solutions
¬A B ¬C, ¬D, ¬E, ¬F
A, ¬B, C, D, E, F
The two solutions are opposites so there is no consistent truth


Overall
Spoiler:
As has been said above the difference stems from the different truth tables of A if D and A and D

A and D is only true if A and D are true
A if D is only false if A is false and D is true.

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Re: Another six-statements logic puzzle

Postby TheGrammarBolshevik » Thu May 07, 2015 10:47 pm UTC

Kingreaper wrote:
Spoiler:
Both of these are consistent, until you look at F again; F says that if E is true then A is true.

But there is a possible solution (aBcdEf) wherein E is true and A isn't. This proves that F is false, and means that aBcdEf is the only possible solution.

Spoiler:
I thought that the "if" clause was meant to be interpreted as a material conditional.
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Kingreaper
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Re: Another six-statements logic puzzle

Postby Kingreaper » Fri May 08, 2015 1:36 pm UTC

Cradarc wrote:Kingreaper,
Spoiler:
Kingreaper wrote:But there is a possible solution (aBcdEf) wherein E is true and A isn't. This proves that F is false, and means that aBcdEf is the only possible solution.

Notice that f is false in that solution, so aren't you basically saying "f must be false because there is a consistent solution in where it is false"?

Suppose you assign truth values to every statement and feed it into a computer. The computer will not search for all the solutions, but merely check to see if your submitted answer is a solution. If you feed it AbCDEF, what will be the response?

Spoiler:
No, I'm not just saying "F must be false because there is a consistent solution in which it is false".

I'm saying "F is false because 'A is true if E is true' is false". It seems to me that F is as false as if it said "Bananas are spherical".

Such a computer would, assuming it was doing a thorough check, notice that "A is true if E is true" is false, and therefore reject AbCDEF, because F is false.

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SPACKlick
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Re: Another six-statements logic puzzle

Postby SPACKlick » Fri May 08, 2015 1:45 pm UTC

Kingreaper wrote:
Cradarc wrote:Kingreaper,
Spoiler:
Kingreaper wrote:But there is a possible solution (aBcdEf) wherein E is true and A isn't. This proves that F is false, and means that aBcdEf is the only possible solution.

Notice that f is false in that solution, so aren't you basically saying "f must be false because there is a consistent solution in where it is false"?

Suppose you assign truth values to every statement and feed it into a computer. The computer will not search for all the solutions, but merely check to see if your submitted answer is a solution. If you feed it AbCDEF, what will be the response?

Spoiler:
No, I'm not just saying "F must be false because there is a consistent solution in which it is false".

I'm saying "F is false because 'A is true if E is true' is false".

Such a computer would, assuming it was doing a thorough check, notice that "A is true if E is true" is false, and therefore reject AbCDEF, because F is false.
Spoiler:
You seem to be misunderstanding the conditional form. A is true if E is true (more often formulated If E is true then A is true) doesn't mean E makes A true it means that the truth value of A coincides with the Truth value of E such that IN THIS CASE it is not the case that E is true and A is false. It does not mean that there is no such case.


I also have a 6 statement version I quite like because of how forumlaic it is
A. B and C are both true
B. C or D is true
C. Not both of D and E are true
D. Neither E nor F is true
E. One of F and A is true
F A is true if and only if B is true

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Wildcard
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Re: Another six-statements logic puzzle

Postby Wildcard » Sun May 10, 2015 6:24 am UTC

SPACKlick wrote:I also have a 6 statement version I quite like because of how formulaic it is.
A. B AND C = true
B. C OR D = true
C. D NAND E = true
D. E NOR F = true
E. F XOR A = true
F. A XNOR B = true
FTFY. And yes, I like its forumlaicness. But am too lazy to solve at the moment. :)
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SPACKlick
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Re: Another six-statements logic puzzle

Postby SPACKlick » Sun May 10, 2015 11:17 am UTC

I'd have framed F as A IFF B but potato potahto

Nitrodon
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Re: Another six-statements logic puzzle

Postby Nitrodon » Sun May 10, 2015 3:56 pm UTC

Solution to this new version:
Spoiler:
If C is false, then D and E are both true. However, E (and F) must be false if D is true, so this is a contradiction. Hence C must be true. B must be true because C is, and from here we can simply evaluate A, F, E, and D in that order.

A: true
B: true
C: true
D: false
E: false
F: true


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