Removing a unit square from a $2\times 2$ square we get a piece called L-tromino. From the fourth line of a $7 \times 7$ cheesboard some unit squares have been removed. The resulting chessboard is cut in L-trominos. Determine the number and location of the removed squares.
Problem
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Tags: combinatorics
socrates
30.05.2015 19:09
Bump! Bump!
sirknightingfail
31.05.2015 09:10
I'm pretty sure any location in the 4th line can be removed, if there is only one square removed.
Also, note that there must be exactly 4 L-trominoes that are in the 1st row, and that then leaves two unused squares in the 2nd row. It's possible to prove with casework that the number of L -trominoes that occupy both row 3 and 4 is at least 3.
By symmetry, this forces there to be at least 6 L-trominoes that have a piece in the 7th row. Therefore, there must be at most one unoccupied square in the 7th row. Finding configurations for this is left to the reader.