A $6 \times 6$ board is given such that each unit square is either red or green. It is known that there are no $4$ adjacent unit squares of the same color in a horizontal, vertical, or diagonal line. A $2 \times 2$ subsquare of the board is chesslike if it has one red and one green diagonal. Find the maximal possible number of chesslike squares on the board. Proposed by Nikola Velov
Problem
Source: 3rd Memorial Mathematical Competition "Aleksandar Blazhevski - Cane"- Junior D1 P2/ Senior D1 P1
Tags: combinatorics, board, Coloring, chesslike, maximum
Jalil_Huseynov
17.01.2022 21:53
Since I can't add more than 3 attachements, I will post my solution in 2 posts.
The answer is $20$.
See the second picture.
square I mean 2x2 square
. Case 1:Square which is on center of $6\times 6$ board is chesslike. Then see the third picture. At most $1$ of same colored squares can be chesslike. Also see the squares that I colored in the first picture. Also at least $1$ of them isn't chesslike. Since all squares in picture $1$ and picture $3$ are different we get we have at least $5$ square that aren't chesslike.
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Jalil_Huseynov
17.01.2022 22:01
Case 2: The senter square isn't chesslike. Then focus on $4$ squares that have $2$ common cells with center square. Observe that at least $2$ of these $4$ squares aren't chesslike. So now we have at least $3$ square which aren't chesslike. Now see the picture. At least $1$ of squares that are colored in the same color isn't chesslike. So we get at least $2$ additional squares which aren't chesslike. So from both cases we get that there can be at most $20$ chesslike squares. So we are done!
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