# Tiling a 4 X 11 board. [closed]

Prove that a 4 x 11 rectangle cannot be tiled by L-shaped tetrominoes.

• I have no idea how to start the proof. Commented Jul 25, 2019 at 4:29
• I can't resist linking to this (mildly embarrassing for me) old thread. Commented Jul 25, 2019 at 4:48
• But the question is in need of a bit of context as outlined in our guide fore new askers :-( Commented Jul 25, 2019 at 4:50

I am considering a grid with $$4$$ column and $$11$$ rows. Now, color the first and the third row with black color and the second and fourth with white. Clearly, whenever you place a $$L$$-Shaped domino there, it will cover either $$(\hbox{3 black position and 1 white})$$ or $$(\hbox{1 black and 3 white})$$. However, the number of black and white position is same in the hole grid. Thus, $$L$$-Shaped domino must exist in pairs. To be more precise, for every domino covering $$3$$ black and $$1$$ white space, there must be another domino covering $$1$$ black and $$3$$ white positions. Thus, total number of $$L$$-Shaped domino must be even. But each $$L$$-Shaped domino has $$4$$ blocks in it. Thus total number of blocks that an even number of dominoes can cover will be a multiple of $$4\times 2=8$$. But total number of blocks in the grid is $$44$$ which is not a multiple of $$8$$. Hence, a $$4\times 11$$ board cannot be covered with $$L$$-shaped dominoes.