Are puzzles that can be solved by translating statements...

Are puzzles that can be solved by translating statements into logical expressions and reasoning from these expressions using truth tables. Four friends have been identified as suspects for an unauthorized access into a computer system. They have made statements to the investigating authorities. Alice said "Carlos did it." John said "I did not do it." Carlos said "Diana did it." Diana said "Carlos lied when he said that I did it." a) If the authorities also know that exactly one of the four suspects is telling the truth, who did it? Explain your reasoning. b) If the authorities also know that exactly one is lying, who did it? Explain your reasoning.

Are puzzles that can be solved by translating statements into logical expressions and reasoning from these expressions using truth tables.
Four friends have been identified as suspects for an unauthorized access into a computer system. They have made statements to the investigating authorities. Alice said "Carlos did it." John said "I did not do it." Carlos said "Diana did it." Diana said "Carlos lied when he said that I did it."
a) If the authorities also know that exactly one of the four suspects is telling the truth, who did it? Explain your reasoning.
b) If the authorities also know that exactly one is lying, who did it? Explain your reasoning.

Key Concepts

Logical Expressions

This concept involves translating natural language statements into formal logical statements or expressions. In puzzles, this helps to clearly define the relationships and conditions described in a problem, enabling systematic analysis and minimizing ambiguity in reasoning.

Truth Tables

Truth tables are a method used in logic to systematically list all possible truth values for logical expressions. They are essential for evaluating which combinations of truth values satisfy the given conditions in a puzzle, especially when dealing with multiple statements and constraints.

Deductive Reasoning

Deductive reasoning involves drawing specific conclusions from general premises or conditions. In logical puzzles, it is used to eliminate possibilities that do not meet the problem’s constraints, such as the exact number of truth-tellers or liars, thereby pinpointing the correct solution.

Constraint Analysis

Constraint analysis is the process of applying and testing the conditions provided in a problem, like 'exactly one person is telling the truth' or 'exactly one person is lying.' This approach is crucial in narrowing down the possible scenarios and ensuring that the final conclusion is consistent with all given restrictions.

Transcript

00:01 All right, so four friends have been identified as suspects.

00:04 They've made statements responding to their authorities.

00:08 Alice has said that carlos did it, so i'm going to see.

00:16 Alice says that carlos did it, so i'm going to write that as access of carlos.

00:19 Carlos is the one who did the access.

00:22 John said i did not do it, so john is saying that he did not do it.

00:31 John.

00:33 Carlos said deanna did it.

00:35 Carlos is saying deanna did it, and deanna is saying carlos lied when he said that i did it.

00:43 So deanna said that carlos did that.

00:48 Carlos lied.

00:51 I'm just going to put that as not carlos.

00:54 The authorities know that exactly one of the four suspects is telling the truth.

00:58 Then who is it that did it? let's see.

01:02 If we assume that alice is telling the truth, then we conclude that carlos did it, of course.

01:11 Then we have, of course, not john, not carlos, not deanna.

01:20 Not john means that john did it because he's lying, right? but that is already a problem, right? if only one person is...

01:36 If we're assuming here that only one person actually did the access, then alice can't be telling the truth because then john would have to be lying.

01:45 So that doesn't work at all.

01:47 Let's assume instead that alice is lying, as are carlos and deanna, and john is telling the truth.

01:57 From this, we conclude that carlos did not do it.

02:02 John telling the truth, we're saying that john did not do it.

02:08 Carlos here is lying, so deanna did not do it.

02:13 And deanna is saying that carlos lied, which means not carlos, and that does say that deanna did do it.

02:26 Or rather, deanna is saying that carlos lied, which means not that deanna did not do it.

02:35 Now, on its face, this seems totally normal, right? this seems consistent.

02:41 However, what we're noticing here is that carlos lied, right? so when deanna said that carlos lied, she was telling the truth, and we already assumed that john was the only one who told the truth.

02:52 So this can't be the case.

02:57 Let's go one more here.

02:59 We can see that alice and john both lied.

03:02 Deanna also lied.

03:05 Carlos is telling the truth...

Please give Ace some feedback