Combinatorial clarifications

A few questions have come up about the combinatorics assignment, and here are some clarifications:

• Q3: The colours referred to are the labels (that is, there are three colours 1, 2 and 3). The rectangles referred to really do have four corners (that is single cells, or parts of a single row or column don't count -- or if you like they don't "have all four of their corners the same colour" since they don't have four corner cells).
  
• Q4: The absolute value bars refer to the size of the set (in general absolute value bars should represent the size of something).
• Q4: The question defines a number for each i, which is the number of black things to its right plus the number of white things to its left (thinking of the numbers from 1 through 2005 arranged from left to right in the usual way). If it should happen that the colouring is such that exactly one of these numbers is odd, then the problem is to determine what that odd number could be.
• Q5: If a switch is used it changes the state of both lamps which it is attached to. That is, if they were originally in different states they will still be in different states (but the opposite ones!)
• Q6: The circumscribing rectangle of a pair of points (at least in this question) has its edges parallel to the coordinate axes and has the points at diagonally opposite corners.