Hernan wants to paint a $8\times 8$ board such that every square is painted with blue or red. Also wants to every $3\times 3$ subsquare have exactly $a$ blue squares and every $2\times 4$ or $4\times 2$ rectangle have exactly $b$ blue squares. Find all couples $(a,b)$ such that Hernan can do the required.
Problem
Source: Mathematics Regional Olympiad of Mexico Southeast 2021 P4
Tags: combinatorics, board, Mexico
e61442289
23.10.2021 04:18
AlanLG wrote: Hernan wants to paint a $8\times 8$ board such that every square is painted with blue or red. Also wants to every $3\times 3$ subsquare have exactly $a$ blue squares and every $2\times 4$ or $4\times 2$ rectangle have exactly $b$ blue squares. Find all couples $(a,b)$ such that Hernan can do the required. somebody?
it just satisfies $(a, b) = (9, 8)$, not sure though
Justpassingby
06.01.2022 22:55
e61442289 wrote:
it just satisfies $(a, b) = (9, 8)$, not sure though
Almost.
The answer is $(a,b)=(0,0)$ and $(a,b)=(9,8)$, which can be achieved by coloring the board all red and coloring the board all blue, respectively. Now let's see that these are the only possibilities.
Consider a $6\times 6$ sub-board and place 4 $3\times 3$ squares as shown in the figure, as well as 4 $2\times 4$ and $4\times 2$ rectangles as shown in the next figure.
Let $x$ be the number of blue squares in the central (green) $2\times 2$ square. We must have $x=4a-4b$ so $x$ is a multiple of 4 and at the same time $0\le x\le 4$, so $x\in\{0,4\}$ and $x$ is a fixed number for all $6\times 6$ sub-boards. Then, all green squares in the next figure must either be all red or all blue, from which the result follows.