Let $2n\times 2n $ board. I cover it with dominoes $1\times 2$ s.t. every cell is adjacent exactly one cell coverd by a domino. I have to find the maximal number of dominoes that can be placed in this way.

I think that the number is $\frac { n\cdot (n+1)}{2} $. I succeed for $n\in \lbrace 1, 2, 3 \rbrace $ and I try to prove it by induction.

It's right my answer?

2 cells are adjacent if they share a common side

  • 3
    $\begingroup$ What do you mean by "every cell is ad[j]acent exactly one cell cover[e]d by a domino"? $\endgroup$ – Dr. Mathva Apr 9 at 10:12
  • $\begingroup$ @Dr. Mathva I added this information $\endgroup$ – Problemsolving Apr 9 at 10:26
  • $\begingroup$ It may be easier to solve the problem by induction on a $2n\times2m$ board instead. $\endgroup$ – P. Quinton Apr 9 at 10:39
  • $\begingroup$ @P. Quinton It's my answer right? $\endgroup$ – Problemsolving Apr 9 at 10:47
  • $\begingroup$ I think that it's is problem 2 form EGMO 2019 (first contest was yesterday). See here artofproblemsolving.com/community/… $\endgroup$ – richrow Apr 10 at 18:50

To show it is possible to place $n(n+1)/2$ dominoes, generalize the domino placements below.

enter image description here

To prove this is optimal, given a placement of dominoes, let

  • $x$ be the number of interior dominoes,
  • $y$ be the number of dominoes with both halves on the border,
  • $z$ be the number of dominoes with one half on the border and the other in the interior.

Note that a perimeter domino covers $4$ perimeter squares, and a half-interior domino covers at least three perimeter squares. Since there are $4(2n-1)$ perimeter squares total, we must have $$ 4y+3z\le 4(2n-1). $$ Furthermore, an interior domino covers $8$ squares, a border domino not on the corner covers $6$ squares, an a half-interior domino covers $7$ squares. There are at most $4$ corner dominoes, which each cover one fewer square. Therefore, $$ 8x+6y+7z\le 4n^2+4 $$ Putting this all together, we get \begin{align} x+y+z &=\frac18(8x+6y+7z)+\frac1{16}(4y+7z)-\frac{5}{16}z \\&\le \frac18(4n^2+4)+\frac1{16}(4(2n-1)) \\&= \frac12n(n+1)+\frac{1}{4} \end{align} Finally, since $x+y+z$ is an integer, we further conclude that $$x+y+z\le \left\lfloor\frac12n(n+1)+\frac14\right\rfloor=\frac12n(n+1).$$


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