Chess board problem

Chess board problem

Postby Guest » Thu Oct 04, 2018 6:49 pm

I need help with a problem:
Chess board 2018x 2018 was covered with one square plate 2x2 and [tex]\frac{2018^{2} - 4}{5}[/tex] rectangole plates 1x5 in the way that every field is covered by exactly one plate (you can rotate plates) You have to prove that plate 2x2 DO NOT COVER any of the fields on the edge of board.

Please help me, i dont have concept how to start

Re: Chess board problem

Postby Guest » Fri Oct 05, 2018 4:16 am

The clue is that its a chessboard, chessboards have a specific coloring of its squares which can be used to solve tiling problems like the mutilated chessboard problem. Look up its solution, figure out how it works and adapt it to your problem

They use 2 colors for 1 by 2 tiles, but you have 1 by 5 tiles so maybe you need more colors. Also does it matter how you color, what if you change the pattern?

By the way if the bottom left corner is (0,0) so the center of the bottom left square is (0.5, 0.5) then the 2 by 2 tile must be placed so its center is at (4+5n,4+5m) for integers n,m. Its easy to check you can always find a tiling for these configurations.

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