# Squares and Straight Tetrominoes

Here is a problem from the 2009-2010 International Internet Mathematics Olympiad run by the Ariel University Center of Samaria (Israel).

Workers were supposed to cover the floor of a rectangular room with tiles of two different sizes: 2×2 and 1×4 (the area of the floor is such that it can be covered completely by a certain combination of the two kinds of tiles). The necessary amount of tiles was obtained, but during their transportation to the destination three tiles of the size 2×2 were broken. However, there were three spare 1×4 tiles. Prove that it is now impossible to cover the entire area of the floor with the available tiles. (Cutting tiles is not allowed.)

Solution

Workers were supposed to cover the floor of a rectangular room with tiles of two different sizes: 2×2 and 1×4 (the area of the floor is such that it can be covered completely by a certain combination of the two kinds of tiles). The necessary amount of tiles was obtained, but during their transportation to the destination three tiles of the size 2×2 were broken. However, there were three spare 1×4 tiles. Prove that it is now impossible to cover the entire area of the floor with the available tiles. (Cutting tiles is not allowed.)

### Solution

Imagine the floor covered by a rectangular grid of 1×squares. Color each odd grid square in every odd row:

It should be clear that every 2×2 square, align with a grid, covers exactly one painted 1×1 square. On the other hand, a 1×4 pieces always covers an even number of painted squares. Three broken squares would have covered 3 - an odd number - of painted squares, something the 1×4 pieces could not do.

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