The question is

You have a chessboard (8 × 8) plus a big box of dominoes (each 2 × 1). I use a marker pen to put an “X” in the squares at two locations. These two locations correspond to a black and white square, not necessarily adjacent. Is it possible to cover the remaining 62 squares using the dominoes without any of them sticking out over the edge of the board and without any of them overlapping? You must not damage the board or the dominoes in the process or do anything weird like standing them on their ends—just answer the question

I'm confused with the provided solution:

Imagine a closed path on the chessboard that passes through every square exactly once (moving horizontally and vertically, eventually returning to the original square). The two “X”s, unless adjacent, divide this path into two sections. Since one “X” is on black, and one is on white, the two sections each cover an even number of squares. They may thus be tiled using the dominoes. If the two “X”s are adjacent, the solution is obvious

I don't understand the part about the 2 X's dividing the path into 2 sections. For simplicity, we can consider the bottom 2 corners of the chess board as having the X's. Where and how does the division occur? What does it look like?


Consider the following closed path where the black dots represents the two chosen squares.

enter image description here

Do you see the two sections in which the path is divided by the two squares? Any couple of squares of different colours works!

Any CLOSED path along the 64 squares can be used. This is another one:

enter image description here

  • $\begingroup$ Ah I see. But in general, can we always find a path that can be divided into 2? $\endgroup$ – student010101 Oct 11 '20 at 19:30
  • $\begingroup$ Given this closed path you can take any couple of squares of different colors. $\endgroup$ – Robert Z Oct 11 '20 at 19:32
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    $\begingroup$ Oh. I was thinking you made this path based on how the black dots were situated. But you actually made the path first and then placed two black dots. $\endgroup$ – student010101 Oct 11 '20 at 19:33
  • $\begingroup$ @student010101 Yes, exactly. $\endgroup$ – Robert Z Oct 11 '20 at 19:36
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    $\begingroup$ @Dr.Mathva Any CLOSED path along the 64 squares can be used. See my edit. You can try to find a different one. $\endgroup$ – Robert Z Oct 12 '20 at 7:27

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