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Suppose you have a grid of squares that has even dimensions, with at least one dimension greater than or equal to 4 squares, and from one corner you remove a 1x4 rectangle of those squares

for example:

□□□□□□
□□□□□□
□□□□□□
XXXX□□

Can you fill in that grid using as many copies of the following shapes as you like?

(Each shape can be rotated any of the four ways, and can be flipped/mirrored) 

□□
□□

□□
 □□

□□□□
  □□□□

If you can, provide an example solution. If you cannot, then you should provide a reasonable argument to why it can't be done.

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No, you cannot:

Color the grid like this.

enter image description here
Since the grid (before removal of the four cells) has even dimensions, it is made up of 2x2 blocks with each color once. So each color appears the same number of times.
All of the given shapes will always cover the same number of squares of each color: the first two cover one of each, and the last covers two of each. But after marking off the unused cells, the grid has more red and blue cells than yellow and green. So you can't cover the grid perfectly.

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  • $\begingroup$ that’s pretty much my reasoning! $\endgroup$
    – micsthepick
    May 12, 2019 at 1:55
  • $\begingroup$ Danf Deus actually gave an answer! For the first time in forever Okay in all seriousness, this is pretty amazing. Also the first post by micsthepicks I've seenin a while. Good to see you again @micsthepick $\endgroup$
    – Prince North Læraðr
    May 12, 2019 at 5:03

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