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- Thread starter LifeIn
- Start date

Treating this scenario similar to a standard "

The array of dominoes where each domino covers two squares each - transforms a square matrix of the original 6 x 6 array into a 3 x 6 array or a 6 x 3 array of dominoes. [ using the standard row, column definition ]

The vertical grid in the 3 x 6 array:

- Counting the spans between the domino columns instead of the domino columns themselves leaves 6 - 1 = 5 spans.

- Because there are an odd number of spans between the domino columns, the third span (counting from the right or the left side) must bisect the matrix through the center of the array.

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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.

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.

Thanks -

Their answer:

"3. There are five horizontal lines and five vertical lines that can cut the board. Since the area on either side of such a line is even, each of these ten lines must cut an even number of dominoes. No domino is cut by more than one line. For each line to cut a domino (and hence cut at least one other domino), we would need at least 20 dominoes.

Since there are only 18, some line must cut the board but not cut any of the dominoes."

Yes, since there are 3*6 = 18 dominoes (or 6*3 = 18 dominoes) there is at least one span between the array that "cuts" the array.

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