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Saw this query within the e-book, “A Moscow Math circle” by Dorichenko.
Eighteen 2×1 dominoes cowl a 6×6 board with out overlapping one another or the edges of the board.
Prove that, for any such association, it’s doable to chop the board into two items alongside a vertical or horizontal line with out chopping a single domino.
First and extra necessary query : Can you please assist me full my reply to the above query ? This is my strategy:
Let us label the rows as a,b,c,d,e,f and columns as 1,2,3,4,5,6.
Consider an association the place there is no such thing as a domino occupying any 2 adjoining horizonal squares. For occasion, allow us to say that there is no such thing as a domino that occupies each 3 and 4. Then, on this association, we will merely draw a vertical line between 3 and 4 with out chopping any domino.
Similarly, contemplate an association the place there is no such thing as a domino which is occupying two adjoining vertical squares, say ‘a’ and ‘b’. Then we will merely draw a horizontal line between ‘a’ and ‘b’ with out chopping any domino.
Therefore, the one strategy to not have the ability to lower the board with out chopping a domino/ dominos is when there’s no less than one domino between each two adjoining horizontal bins of the board i.e there’s no less than one domino every on 1-2, 2-3, 3-4, 4-5 and on 5-6.
And additionally, there’s no less than one domino between each two adjoining vertical squares i.e there’s no less than one domino between a-b , b-c , c-d, d-e and e-fNow, how will we show that such an association isn’t doable ?
Question 2: How would you’ve gotten solved the query ?
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