(Note: Hover over underlined words only if you need further explanations after having read through the argument in its entirety.)
The claim of physicalism is that sentient beings can be explained completely in physical terms. The following argument describes a scenario in which you have access to a piece of information that is expressed nowhere in the physical universe. Accessing this piece of information requires your sentience, which you should invoke while reading, by doing the imagining in the first person.
Suppose the physical universe was in the following state: It has a red room and a blue room. Inside each of the rooms, is a human. The two humans are physically identical. The humans know about the experiment.
Now suppose you are one of these humans. So the human in the other room is a copy of your physical body, starting off in the same state as your physical body.
There are two such cases possible:
If we look at these two cases of the universe from any third person perspective, they are physically indistinguishable. Even an omniscient observer of the physical universe would not be able to distinguish between the cases. From a first-person perspective, however, you are able to distinguish between the two cases (by the color of the room you find yourself in). So you have access to a fact that an omniscient observer of the physical universe would not have access to.
If you have access to information that is not expressed in the physical universe, then you are a counterexample to physicalism.