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[Jreengus]

If you want a hint:

Spoiler (click to show/hide)

The first prisoner has a 50/50 chance everyone else survives. They have to agree on a binary piece of information the first prisoner can transmit that will allow everyone else to guess their number based on the information available to them.

[Leafsnail]

Isn't that what I suggested?

[Darvi]

He made a rule that says it won't count.

[Jreengus]

The original question may have been worded badly. Assume that rather than saying their colour the warden holds a hat of each colour and they point to the colour they believe they have and the warden calls it out.

The point is the only information they can transmit to the next person is whether they chose red or blue.

[Simmura McCrea]

They choose the hat colour of the next person?

[Darvi]

Well no, 'cause then their hat might be different and then they die.

Unless you suggest what leafsnail said.

[Leafsnail]

I'm grasping at several things that feel close.  I have a strong feeling it's to do with the number of hats the first prisoner can see ("Can you see more reds or more blues?" works if prisoner 2 can see an equal number of each colour of hat).

[G-Flex]

You can't assume that there's an equal amount of blue hats as there is red hats, though. You're describing a scenario that you can't really count on.

[Leafsnail]

...No, wait.

All prisoner 1 needs to say is whether he sees an odd or even number of red (or blue) hats.  Let's say he says "Red" for odd and "Blue" for even.

Prisoner 1's hat is irrelevant to everyone else, so we'll discount it.  Out of the remaining nineteen, EVERY prisoner will know all the hats in front of and behind him.  From them, and the information that the total number of red hats is odd or even, he can work out what colour his own hat is.

A more graceful question for prisoner 1 to answer which would have the same effect is "Which colour can you see an even number of?"

[G-Flex]

Oh, right, I see what you mean. Yeah, that should work.

[Jreengus]

And leafsnail gets it!

[Leafsnail]

That's a pretty good one, actually.

Although in real life, that'd only be the start of the story.  As prisoners, you're likely not to be hugely numerate or gifted with amazing memories.  As for who exactly gets the role of being Prisoner 1...

[Darvi]

The warden chooses him arbitrarily of course. Poor nr1.

'cept for the fact that he's a convict, but whatev'.

[Leafsnail]

Ah, even worse.  Now you have a guy who possibly can't count in the crucial role...

Although if the prisoners got to choose, everyone would suddenly and mysteriously forget how to count.

[Darvi]

It really depends on whether they are social or not. Whoever gets picked has to make a sacrifice if unlucky.

Of course, even if they weren't social, they'd still lynch arseholes, so there's motivation for ya.