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As a birthday present last year, I received some fridge magnets. They didn't come as a puzzle, so I don't know if they have a solution, but I made a puzzle out of them anyway.

The magnets are tetrominoes. There are 7 of each shape. Is it possible to arrange them into a 7x28 rectangle so that they are all used and all inside the rectangle?

The closest I have managed is this:

enter image description here

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    $\begingroup$ As an aside, just because it's a puzzle doesn't mean it has a solution; the 15 puzzle, for example, had a popularized unsolvable configuration. $\endgroup$
    – user1502
    Commented May 28, 2015 at 8:58
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    $\begingroup$ If you want a solvable challenge, use these pieces to build a 12x16 rectangle (there will be one left over). $\endgroup$
    – Mike Earnest
    Commented May 28, 2015 at 14:22
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    $\begingroup$ Thank you to the upvoters! This question has just become my biggest SE achievement as 'the highest voted question on the site' $\endgroup$
    – James Webster
    Commented May 29, 2015 at 7:23
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    $\begingroup$ en.wikipedia.org/wiki/… $\endgroup$
    – RobEarl
    Commented May 29, 2015 at 14:21
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    $\begingroup$ You've got an odd number of "T"s, so no, not possible.... Oh, I see @Tryth already go it. $\endgroup$
    – RBarryYoung
    Commented May 29, 2015 at 15:06

1 Answer 1

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It is impossible.

Let the $7\times 28$ area be painted with black and white squares in a checkerboard pattern. Every piece will cover $2$ black and $2$ white squares, except the T-piece, which covers $3$ of one color and $1$ of another. Since there are $7$ T-pieces, a tiling that uses every piece cannot cover the same number of black and white squares. Since the board contains the same number of black and white squares, it is impossible.

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    $\begingroup$ Nice! Also shows that you can't cover a 4x49 or 14x14 board. $\endgroup$
    – Mike Earnest
    Commented May 27, 2015 at 7:08
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    $\begingroup$ Very reminiscent of the classic "dominoes on a chess board with the corners removed" proof. I like it. $\endgroup$
    – undergroundmonorail
    Commented May 27, 2015 at 12:56
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    $\begingroup$ @undergroundmonorail is referring to the Mutilated Chessboard Problem (en.wikipedia.org/wiki/Mutilated_chessboard_problem). It's not all corners which are removed, just the 2 opposing (catty-corner) corners (which are the same color). The solution hinges on there being the same number of white boxes as there are black, as does this proof. $\endgroup$
    – dberm22
    Commented May 27, 2015 at 14:46
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    $\begingroup$ @Zibbobz: Sure. Once you can make one rectangle with $n=4$ you can just line up as many of those rectangles as you want. $\endgroup$
    – Ross Millikan
    Commented May 27, 2015 at 16:40
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    $\begingroup$ @kangacam How could the T piece ever do anything other than 3 and 1? It's a single square (the center of the T piece) with three squares all adjacent to that one single square. Thus the single square in its center must always be a different color than the three other squares by definition. The zigzag pieces on the other hand must cover 2 and 2: their center two pieces are always opposite colors, and each has one other piece adjacent which also must be an opposite color. $\endgroup$
    – Joe
    Commented May 28, 2015 at 15:44

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