A question on a hint in Clocktower puzzle (spolers, obviously)

[lentinant]

So I did get into the Clocktower puzzle secret room. I understand the rules of the puzzle, but I don't get the third clock hint. My issue is that it is not exactly a statement, but an action. "Set this clock to have all different numbers", but it is red, so what exactly does it mean? It can't mean "Set all numbers to be identical", as the due to another hint, it must contain 7, and obviously there could be no 7:77.

So, am I correct in understanding this as "the clock must have at least some identical numbers"? By that clue, and by other restrictions, the result would be 5:57.

[Silyon]

This sounds correct. Recall that just like in the parlor game, a false statement does not have to mean the complete opposite is true. You're merely looking for a condition under which the statement isn't true. Given other restrictions, 5:57 is the ONLY time that works for that clock while leaving the memo false.

[lentinant]

Thanks for confirming it. I do seem to have some logical fallacy in my thoughts, as things do not align.
So, the rules are:

• All clocks must have different time.
  
• 5 of 8 clocks must have 00 minutes
  
• Clock #3 should have at least some identical numbers and a 7
  
• Clock #5 should have 00 minutes, and a 7.
  
• No clock can have 1, 2, 3 and 4 in their time.
  
• Clock #7 has a time similar to another clock, but in reverse.
  
• Clocks are sorted in ascending order (the note is red, but it is also hand-written).

So, here are my conclusions so far.

• Clock #5 is 7:00, that's pretty easy to figure out.
  
• As per first post, clock #3 seems to be 5:57.
  
• Clock #7 seems to not have 00 minutes, as no clocks can start with 0. So this is second of three clocks that doesn't have 00 minutes.
  
• Clocks can only start with 5, and considering third clock is 5:57, clock #1 can only be 5:00.
  
• Clock #2 is between 5:00 and 5:57, so I assume it is a last clock that should not have 00 minutes.
  
• Clock #4 is between 5:57 and 7:00, so I assume it is 6:00. By the same logic, clock #6 seems to be 8:00.

This is where I encounter the issue. What is clock #7? I would assume it is inverted time of clock #2 or clock #3, BUT, it needs to start AT LEAST with 8. So, it can't be inverse of #3, which would be 7:55. But that means it has to be inverse #2. Which means, #2 has to end with 8, but with other rules, that would be 5:58 or so, which is not possible, as it has to be lower than 5:57.

I'm not sure where exactly my conclusions have started to be incorrect.

[lentinant]

Ah, is clock #2 5:08, which would make clock #7 8:05? I've forgot the rules do not rule out 0.

[Silyon]

Originally posted by lentinant:

Ah, is clock #2 5:08, which would make clock #7 8:05? I've forgot the rules do not rule out 0.

Got it before I could reply, good job. Sometimes all that's needed is to go over the established rules once more, I suppose.