# Can Igor trust the message?

After deciphering the mysterious letter Igor's boss received, Igor realized that before he could do anything about it, he'd better find out if the letter is real information, or meant to sabotage his work. His neighbors weren't really fond of their work, which was loud and often took place at night, and produced unfortunate smells.

He knew his neighbors wouldn't just give him the answers he wanted, but he set out with a plan. He gathered some of his loose electronics and put together what he needed, then visited the houses.

Igor knew for sure the letter was sent by one of three neighbors: Vladimir, Khamen, or Michael. He went to the three houses and got three statements from each neighbor. Here are the statements:

1. Oh, that letter? I sent that letter!
2. And if I did, I put misleading clues in it!
3. Hm. Well, actually, Michael sent the letter.

Khamen:

1. Vladimir? That guy is the worst! If he wrote you that letter he definitely would have put false information.
2. Michael? Wow, he's just as bad! He would have given you really bad information.
3. Eh... well, I would have put the wrong details in the letter too.

Michael:

1. Well, if Khamen had sent that letter, it definitely would have had fake details
2. And, as it turns out, he DID write that letter.
3. Although honestly, it doesn't really matter who wrote it.

Having taken the statements, Igor rushed home. "Ah, the fools! They didn't realize I had brought my device that would be able to tell if they were lying or not! Now all I need to do is... Oh."

As he took out the device, he noticed that it had melted a little by coming in contact with some loose electricity he had been keeping in his pocket.

"That's not really good... can I salvage anything here? Hmmm. Well, it looks like they all told the same number of lies, so at least that should help."

Can Igor trust the information in the letter? And does he know who wrote it?

• I hope this is as interesting as I tried to make it, it was tricky to make and ultimately felt easier than I wanted it to be, but anyway. I think it's at least completely solid, logically. – ColdFrog Jan 20 '18 at 6:45

The letter has true information and we can't find who wrote it.

Case 1:

Vladmir writes with true information.

• Vladmir puts true information in the letter, therefore his statement (2) and (3) are wrong.

• Khamen lies about Vladmir and Michael, i.e. his statements (1) and (2) are wrong.

• And as it turns out Michael lies about Khamen writing letter, so his statement (2) is wrong. And as Vladmir/Michael write true letters and Khamen writes false letters,who writes the letters does matter, so his statement (3) is wrong.
• Case 2:

Khamen writes with true information.

• Vladmir's statement (1) and (3) are wrong, so Khamen wrote the letter.

• Khamen lies about himself and Michael, i.e. his statements (2) and (3) are wrong.

• And as it turns out Michael lies about Khamen writing fasle letter, so his statement (1) is wrong. And as Khamen/Michael write true letters and Vladmir writes false letters,who writes the letters does matter, so his statement (3) is wrong.
• Case 3:

Michael writes with true information.

• Vladmir's statement (1) and (2) are wrong, so Michael wrote the letter.

• Khamen lies about Michael and Vladmir, i.e. his statements (1) and (2) are wrong.

• And as it turns out Michael lies about Khamen writing letter, so his statement (2) is wrong. And as Vladmir/Michael write true letters and Khamen writes false letters,who writes the letters does matter, so his statement (3) is wrong.
• Are you sure about who wrote the letter? – ColdFrog Jan 20 '18 at 6:49
• I can... not confirm. Although I'm not saying that your logic isn't sound. – ColdFrog Jan 20 '18 at 6:55
• Also, sorry, I realized I didn't include the second important question here. – ColdFrog Jan 20 '18 at 6:56
• Interesting to note that Micheal's 3rd statement is wrong in each individual case but is ultimately correct. – Dan Jenks Mar 6 '18 at 10:31

I disagree with prog_SAHIL's answer. Here's the key difference:

Statements of the form "If X wrote the letter, then Y" are always (vacuously) true if X didn't write the letter. prog_SAHIL sometimes considers these statements to be false.

The letter is accurate, and Michael wrote it.

Reasoning:

Part 1 - how many lies does each person tell:

Everyone makes either 0, 1, 2, or 3 false statements. We can immediately rule out the possibility of 0 false statements, because Vladimir's first and third statements contradict each other. Next, since at most one person wrote the letter, at least two of Khamen's statements must be vacuously true, so at least two of his statements must be true total. Since we already established that not all of the statements can be true, we now know that each person must have made 2 true statements and 1 false one.

Part 2 - is the letter accurate:

All of Khamen's statements are of the form "if X, then the letter is inaccurate". If the letter is inaccurate, all such statements must be true. Since one must be false, we know the letter is accurate.

Part 3 - who wrote the letter:

Vladimir's first and third statements contradict each other, so they can't both be true. Since each person tells only one lie, his second statement must be true. Since we know the letter is accurate, his second statement means that he couldn't have wrote the letter. Thus, his first statement is the lie, so his third statement must be true, meaning that Michael wrote the letter.