Continuity mistake: When we see the Russian guy get shot and he falls out of his chair, he knocks the chessboard off the table. But when Jeffrey Spender shows his team the tape, the guy hits the board so that it spins around, instead of falling off completely. (00:02:40 - 00:07:55)
Continuity mistake: This is probably one of the most intriguing chess plays in the history of mankind. In the first move shown on the screen Gibson moves a white knight from e4 to f6. Then, in the close-up, the situation on the chessboard more or less corresponds to the arrangement of the figures previously shown on the screen. However, in the next shot, when we see the chessboard from the Russian perspective, the arrangement is completely different. Especially, there is no white knight on the chessboard at all. In the following shot the white rook suddenly appears on g8 (and is subsequently beaten by the black king), but in the next move Gibson again moves his white rook on g8, while the black king has miraculously returned to h8 and again beats the white rook. Moreover, Gibson could not play two white rooks, because later on we see that there is still at least one white rook on the chessboard.
Other mistake: In order to leave such bloody footprints, CSM must have stepped into the puddle of blood. It is impossible because the guy shot by CSM could not bleed so much blood in such short time, not mentioning that CSM must have intentionally stepped into it. (00:05:20)
Continuity mistake: The chessboard was knocked down by the falling Russian, however in the next shot it is still on the table. (00:02:55)
Answer: As he stated many times throughout the series, Mulder needed Scully to be sober and skeptical. Whenever Scully's skepticism wavered and she started questioning her own rationality, Mulder would try to restore her sense of skepticism, because he needed her to be clear-thinking.
Charles Austin Miller