Friday, October 14, 2016

checkerboard - Combinatorial Agriculture


You have just acquired a $64$ acre farm, in the shape of a square, and divided into an eight by eight array of one acre subplots. You have $21$ crops to plant. Each crop requires its own $3$ acre plot of land, which must consist of three subplots in the shape of a three by one rectangle. Once the crops have been planted, there will be $64-21\times 3=1$ subplot leftover, where you will build your farmhouse.


It isn't the season for planting yet, so you plan to build your farmhouse now, then plant the crops later. Which subplots can you build your farmhouse on which still allow you to plant all of your crops?



Answer



It is only possible in




$4$ cells



Those cells are



The black ones in the bottom grid. enter image description here



Because



If you tile the grid with three colours, following the above pattern (the right one is symmetrical to the left one), you get 21 green, 21 blue and 22 red cells.

Though, any 3x1 rectangle occupies exactly 1 red, 1 blue and 1 green cell, so the farmhouse must be in a red cell.
Also, it must be in a red cell in both grids (otherwise it would violate the above condition in one grid), and if you do the intersection of red cells (the cells that are red in both grids) you only get the 4 black cells in the bottom grid.



An example of valid configuration is



enter image description here



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