Hmmm, I'm not sure about that. I couldn't get the solution either, until I
looked it up. It from the
1999 Michigan Math Prize Competition.
Since the grid is 6x6, any vertical or horizontal grid line divides the grid into two pieces that have an even number of squares. (6x1, 6x2, 6x3, 6x4, 6x5, 1x6. 2x6, 3x6, 4x6, 5x6 are all even numbers.) Any domino that is cut by one of these lines takes up exactly one square from each of the two pieces that grid line produces. That leaves an odd number of squares in each of the remaining pieces. Since it is impossible to cover an odd number of squares with dominoes (which also cover squares two at a time), any complete covering of the 6x6 grid must have a second domino blocking that grid line so that the two resulting pieces of the grid again have an even number of remaining squares. Therefore of one domino blocks a grid line, then there must be at least two dominoes blocking that grid line. Since there are 10 grid lines (5 vertical and 5 horizontal), that requires 20 dominoes to block them all. But there are only 18 dominoes covering a 6x6 grid. Therefore it is impossible to block all 10 grid lines with a complete covering of a 6x6 grid with dominoes.
Note that the same argument does not work for a 8x8 grid because there are 7 vertical and 7 horizontal grid lines, which require 14 x 2 dominoes (28). An 8x8 grid covered by dominoes uses 32 dominoes. And in fact it is possible to block all 14 grid lines with dominoes in an 8x8 grid.