Math puzzle

I am inspired by a toot of @mx_psi to write down a puzzle I heard last year from a friend.

Three friends are standing around when a demon appears and writes a real number on each one's forehead (they have large foreheads). The demon then invites each of them to write down a finite list of real numbers on a piece of paper. The friends may see each other's foreheads, but must not communicate otherwise. All three simultaneously reveal their papers. (1/2)

Math puzzle

The friends win if any of them writes down the number on their own forehead. If none of them manage to do so, then the demon wins.

To motivate them we may assume that if the friends win then they all receive a small prize such as cupcakes. Otherwise the demon will keep the cupcakes and the friends will not get any.

Under what circumstances can the friends win?

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Math puzzle (hint 1)

Hint 1 comes in the form of an easier puzzle. It is the same as the original but:
- there are only two friends
- the demon writes natural numbers on their foreheads
- they each simultaneously guess a finite list of natural numbers

The friends can always win this game. How?

Math puzzle (Hint 2)

Hint 2: It is independent from ZFC whether the friends can win.

Math puzzle (Hint 3)

Hint 3: The friends can win if (and only if, according to a friend of mine) the Continuum Hypothesis holds.

Math puzzle (Hint 4)

Hint 4: Same game but:
- 2 people
- real numbers
- each person gets to write a countable list

Math puzzle (1% of a solution)

@axiom the first things I'd write down would be the two numbers I can see, just in case. Not sure how to improve my chances from there

Math puzzle (1% of a solution)

@christianp @axiom if each person writes down all the numbers between the two numbers they can see, at least one of them will write their own number.

Math puzzle (1% of a solution)

@loke @christianp Note that they are permitted to write down only a finite list.

Math puzzle (1% of a solution)

@loke @axiom but there are uncountably many of those, unless the numbers on the foreheads are chosen from a countable subset of the reals.

Math puzzle (1% of a solution)

@christianp @loke Being a demon, the demon is free to choose whichever real numbers it desires and is not restricted to any particular subset of the reals.

Math puzzle (hint 1)

@axiom
I get this one, and it suggests that I was on the right track with the original, but I still don't have it...