Solution 41 Let
S be the set of all the students. Denote its cardinality by
α, where
α is an arbitrary ordinal number. Define
a◃b if and only if
a<b.
To each student assign an element, say
ai, of the ordinal
α. Let
F be a good scenario. Then the student
ai answers by guessing that the good scenario is
f(ai). Let
T={ai∈S∣f(ai)=F}. Obviously,
T is the set of those students, who guess their hats incorrectly.
Since
(S,◃) is a strictly increasing order without infinite chains, we know that
T is finite.
Hence the number of students who guess incorrectly will be finite.
Remarks:
We can consider any finite set of colors.
Also, what can we say in the case when
card(S)≥card(R)?