True and false statements

Logic Level 3

Which of the following statements are true and which are false, knowing that the entire set is uncontradictory?

S1. Statements 2 and 3 are either both true either both false.
S2. Exactly one of the statements 4 and 5 is true.
S3. Exactly one of the statements 4 and 6 is true.
S4. Exactly one of the statements 1 and 6 is true.
S5. Statements 1 and 3 are of the same type (both true or both false).
S6. Exactly one statement from statements 2 and 5 is true.

Write the answer as the concatenation of the digits 1 and 0 for the truth values of the statements (true and false) starting from S1 to S6, where for the value true corresponds 1 and for the value false corresponds 0. For example, if the first 2 statements would be true and the rest false, the answer would be 110000.

If the correct answer begins with some number of leading 0s, remove it from writing the answer. For example, if the answer is 001100, write 1100 anyway.

The answer is 10010.

1 solution

Gregory Lewis
Jul 12, 2016

Pick any values for S1 and S2. Use S1 to obtain the value of S3. Check if S5 is true. Check if S6 is true. Use S2 to obtain the value of S4. Check for consistency. Repeat for other values of S1 and S2 until the consistent result is obtained: FTFFTF.

One note, the problem should state to remove any leading zeros, because 010010 was not accepted as a correct solution, but 10010 was.

Thanks for solving it , you are correct.

The problem with the leading 0 is an old one. When I published the problem I inserted the answer with the leading 0 as correct as 010010 which the computer took as 10010 and in that time anyone who would try to solve the problem and insert the answer like 010010 would be marked correct because the computer would not recognize the first 0 and automatically interpret that as 10010. I got such complaints for some other problems which have the similar issue because after some time it seems that computer is capable of readin even the first 0s and mark anyone who enters it like that wrong , thanks for your suggestion , I'll edit the problem accordingly and I hope you like it anyway.

A A - 4 years, 11 months ago

S1 & S5 are of the same form, and since S1 & S3 have been involved twice for both and leaving S5 and S2 behind, therefore S5 & S2 must have the same truth values.

S1 : S2 = S3 { involving S1, S2 & S3 }
S5 : S1 = S3 { involving S1, S5 & S3 }
Both S1 and S3 are neutralised by their own doppelgangers, thus the conclusion is that S5 = S2, whether both are true or both are false.

From our inference above, S6 must be false, and S4 also false (no matter what the truth values of S2 & S5). From both S6 and S4 with double falsehood, we get S1 to be false, the same as S3.

With S3 = S1 = false, we get to return to a true S5 and the twinning true S2.

Answer = 010010

Saya Suka - 3 months ago

This is one example where too much of a positive things isn't always good. I like this puzzle too much that I can't put it down, literally slept on it (on the problem AND on the phone) and accidentally touched the "See the answer / solution" in my sleep. Darn.

Saya Suka - 3 months ago

True and false statements

The answer is 10010.

1 solution

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