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Suppose we have board, of size (16x16)

And 31 (1x4) + 33 (2x2) pieces.

Is it possible to cover up board with those pieces, if so - how?

If not - why?

So far I was unable to think of anything better, then bruteforcing setting by python script, and didnt get result. So I assume it is impossible, or then I made mistake with algorithm.

Any tips welcome

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    $\begingroup$ The answer to these is always no. The usual way to prove it is to find a coloring of the board. You might find a two color one that has the $2 \times 2$ tile covering two of each color and the $4 \times 1$ tile covering $3$ of one color and $1$ of the other. If the board has the same number of squares of each color you are done. You also might find one that the $4 \times 1$ covers four of the same color. No, I don't have one, or I would make an answer. Usually the pattern of colors matches the shape of the pieces. You might also find a pattern using four colors. $\endgroup$ – Ross Millikan Feb 11 '16 at 17:23
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It is impossible. Consider the colouring of the board below.

enter image description here

Suppose it is possible. Then each $2 \times 2$ piece covers exactly one black square and each $1 \times 4$ piece covers either zero or two black squares. But there are $64$ black squares and $64 - 33$ is odd, contradiction.

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