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Christmas is coming and Santa is working hard to wrap all the presents until the 24th of December.

He has six wrapping machines: A, B, C, D, E and F, that work $24/7$ all at the same time. But the harsh weather of the North Pole keeps breaking the machines. Each machine has a panel that indicates the state of one or two other machines. However, if that machine is broken itself, the displayed information on its panel may not be true. However, if a machine is working properly, its panel displays valid information.
The elves made a Main Control Panel so Santa can track all the six individual panels at once. On a certain day, Santa sees the this display on the Main Control:

enter image description here

PROBLEM 01: Given this display, is there any way of having just one broken machine?

PROBLEM 02: If Santa has two broken machines, which ones are they?

This is an old puzzle from a book I had when I was a kid.

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PROBLEM 01

I think this is not possible.
Consider just A and F in isolation.
If both are okay, then at least B and C are broken.
If A is okay, but F is broken, then C and F are broken.
If A is broken but F is okay then A and B are broken.
If both are broken then A and F are broken so we always have at least two broken machines.

PROBLEM 02

In the four scenarios considered in PROBLEM 01, the only one which produces exactly two broken machines is when A and F are both okay, in which case B and C must be the broken machines.

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    $\begingroup$ That was fast! I swear to God, one day I will post a puzzle so hard you will need an entire month to solve it, LOL! That's it. I'm just waiting to accept your answer. $\endgroup$ – Pspl Nov 4 '20 at 10:31
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A compact answer:

A says C is broken. So, one of A and C is broken.
F says B is broken. So, one of B and F is broken.

It is clear from that that at least 2 machines are broken

If 2 machines are broken, they are among A, B, C, F. So E is reliable.
E tells us that between A and C, C is the broken one, and between B and F it is B.

You can check that if only B and C are broken, all other pannels are correct.

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