The player in the center of the room will only give three kinds of requests
to the players on the circle; we call them \(R_1\), \(R_2\), and \(R_3\).
These three requests are defined below and
we also specify the sequence in which these three requests should be given:

The request \(R_1\) asks the players in the chairs \(1\) and \(3\) to change the colors of their hats.

The request \(R_2\) asks the players in the chairs \(1\) and \(2\) to change the colors of their hats.

The request \(R_3\) asks only the player in the chair \(1\) to change his/her hat.
Denote the white hats by \(1\) and black hats by \(1\).
Note that the requests \(R_1\) and \(R_2\) preserve the product of all the numbers,
while the request \(R_3\) changes the product of the numbers. There are \(2^4\) possible configurations of hats.
Half of them have the product equal to \(1\) and half of them have the product equal to \(1\).
Denote by
\(S_+\) the set of all possible configurations that have the product equal to \(1\)
and by
\(S_\) the set of all configurations that have the product equal to \(1\).
We will now define the sequence \(\Sigma_+\) of requests whose terms are \(R_1\) and \(R_2\) only. During the requests in \(\Sigma_+\), the configuration of hats will not move from \(S_+\) to \(S_\) (or vice versa). The sequence \(\Sigma_+\) will be chosen in such a way to guarantee
that at some point all hats become of the same color, provided that the original configuration belongs to \(S_+\). If during the application of the sequence \(\Sigma_+\) the hats never become of the same color, then we are sure that the original configuration is \(S_\). In that case, the player in the center of the room gives the request \(R_3\) and repeats the sequence \(\Sigma_+\). This repeat application of \(\Sigma_+\) will guarantee that at some point the hats all become of the same color. Thus, once we construct \(\Sigma_+\), the desired sequence of operations will be \[\left(\Sigma_+, R_3, \Sigma_+\right).\]
In making \(\Sigma_+\), our goal is to arrange the requests \(R_1\) and \(R_2\) in such a way that if the configuration of hats is in \(S_+\),
then at some point all the hats become of the same color.
We will thus only work with configurations in \(S_+\). It suffices to restrict our attention only to the subset \(S_+^2\) of those configurations that have two terms equal to \(1\) and two terms equal to \(1\).
Configurations in \(S_+\) that are not in \(S_+^2\) are trivial:
the hats are all of the same color and the game is over.
We partition the set \(S_+^2\) in two subsets \(A\) and \(B\).
The subset \(A\) consists of those configurations in which the signs alternate:
\[
A
= \left\{ (1, 1, 1, 1), (1,1,1,1)\right\}.
\]
The set \(B\) consists of the remaining four configurations.
\begin{eqnarray*}
B&=&\left\{(1,1,1,1), (1,1,1,1),\right.\\
&&\left. (1,1,1,1), (1,1,1,1)\right\}.
\end{eqnarray*}
Let us now analyze the request \(R_1\). The request \(R_1\) takes a configuration from \(B\) into a configuration from \(B\). However, \(R_1\) will take a configuration from \(A\) into the one in which all hats are of the same color.
The first term of the sequence \(\Sigma_+\) will be \(R_1\). If the configuration was in \(A\), then the game is over after this request \(R_1\). If the game is not over yet, then we are sure that the configuration is in \(B\).
Assume now that the request \(R_2\) is given and that the configuration of hats is in \(B\). Since the set \(B\) has only \(4\) elements, it is easy to verify directly that \(R_2\) applied to any element in \(B\) will either result in all hats being of the same color, or in all hats falling into a configuration that belongs to \(A\).
Therefore, the sequence \(\Sigma_+\) defined as
\[
\Sigma_+
= \left(R_1, R_2, R_1\right)
\]
guarantees that at some point all hats are of the same color, provided that initially an even number of hats is black.
Thus,
the sequence of requests that, for each initial configuration, guarantees that
the hats will be of the same color at some point is
\[
\left(R_1, R_2, R_1, R_3, R_1, R_2, R_1\right).
\]