Both Raj and Howard discuss the blue shag carpet put down in the nursery, and when the room is seen, in the imaginary scenarios, there is the rug, right in plain sight.
Sheldon explained the Pythagorean Theorem incorrectly. He stated that the square of the hypotenuse is equal to the sum of the squares of the opposite sides and even made motions with his hands demonstrating it incorrectly. The square of the hypotenuse is equal to the sum of the squares of the adjacent sides of a right triangle.
When playing chess, Leonard touches, and even start moving, different pieces in the same turn. In chess, if you touch a piece, then you must move that one and can't switch to another.