Saw this question in the book, "A Moscow Math circle" by Dorichenko.
Eighteen 2x1 dominoes cover a 6x6 board without overlapping each other or the sides of the board.
Prove that, for any such arrangement, it is possible to cut the board into two pieces along a vertical or horizontal line without cutting a single domino.
First and more important question : Can you please help me complete my answer to the above question ? This is my approach:
Let us label the rows as a,b,c,d,e,f and columns as 1,2,3,4,5,6.
Consider an arrangement where there is no domino occupying any 2 adjacent horizonal squares. For instance, let us say that there is no domino that occupies both 3 and 4. Then, in this arrangement, we can simply draw a vertical line between 3 and 4 without cutting any domino.
Similarly, consider an arrangement where there is no domino which is occupying two adjacent vertical squares, say 'a' and 'b'. Then we can simply draw a horizontal line between 'a' and 'b' without cutting any domino.
Therefore, the only way to not be able to cut the board without cutting a domino/ dominos is when there is at least one domino between every two adjacent horizontal boxes of the board i.e there is at least one domino each on 1-2, 2-3, 3-4, 4-5 and on 5-6. And also, there is at least one domino between every two adjacent vertical squares i.e there is at least one domino between a-b , b-c , c-d, d-e and e-f
Now, how do we prove that such an arrangement is not possible ?
Question 2: How would you have solved the question ?