Terrific reasoning! That same "checkerboard coloring" strategy is used a lot for figuring out tiling problems.<p>It is too bad the lamp wasn't made of pentominoes (the 12 Tetris-like pieces with 5 squares vs. your tetrominoes with 4 squares.). See <a href="http://en.wikipedia.org/wiki/Pentomino" rel="nofollow">http://en.wikipedia.org/wiki/Pentomino</a>. There are 2339 ways to form these into a perfect 6x10 rectangle (more if you include rotations and reflections).<p>FYI: The creator of Tetris actually got the idea for his game pieces from Solomon W. Golomb's book "Polyominoes" that introduced all kinds of variations on tiling puzzles and proofs. Chapter one starts using checkerboard reasoning right off the bat. So, you are in good company.
I came here looking for a link to someplace where I can buy this lamp. Since no one has posted such a link yet, here it is:<p><a href="http://www.thinkgeek.com/product/f034/" rel="nofollow">http://www.thinkgeek.com/product/f034/</a><p>(I realize we all have access to Google, but if I save 100 people 10 seconds, then I just saved 1000 seconds!)
Similar to:<p><a href="https://en.wikipedia.org/wiki/Mutilated_chessboard_problem" rel="nofollow">https://en.wikipedia.org/wiki/Mutilated_chessboard_problem</a>
>"Maybe now I can shift my irritation from the lamp itself to whoever designed it to possess such a property. "<p>I wouldn't be mad at the designer. That person designed a lamp and proposed an impossible puzzle that at first glance looked plausible.<p>It is the best way to troll people... ever!
Your friend read Martin Gardner's aha! Insight, which has exactly this proof in it and is a popular book for young mathematical puzzle enthusiasts.
Here's one nice-looking configuration I found. You seem to be implicitly assuming you can only have a depth of one unit but that's possibly an unnecessary limitation. I'm working on seeing if some kind of perfect cuboid can be made.<p><a href="https://3dwarehouse.sketchup.com/model.html?id=u2d94f70c-6825-4fed-bc3d-bfc700d99eec" rel="nofollow">https://3dwarehouse.sketchup.com/model.html?id=u2d94f70c-682...</a> (there's a webgl viewer)
A few things:<p>1) Why limit yourself to 4x7? The 1988 NES version of Tetris is 10 units wide.<p>2) There isn't any malicious design, you simply get 1 of each shape (one of the L pieces in the author's photo is reflected, should be turned the other way).<p>3) In Tetris, a full row is removed immediately so having a complete rectangular shape that occupies the full available width is unrealistic.<p>Pedantry aside, you'd have to ask Alexey Pajitnov if there was any devilry involved in choosing the shapes since the makers of the lamp have faithfully included a full set. Also, I personally prefer the aesthetic of a lamp arranged in such a way as to leave a hole in each row rather than a plain wall of coloured squares.
> <i>Maybe now I can shift my irritation from the lamp itself to whoever designed it to possess such a property.</i><p>There's no "design" involved in the choice of pieces: there are seven different ways to connect four squares in an orthogonal grid, "tetrominoes", assuming you allow for pieces to be rotated but not reflected. Tetris uses all seven, and so does the lamp. (Although the lamp's design apparently allows pieces to be reflected, i.e. rotated outside of the grid, so the S and Z pieces can be considered the same, as can L and J.)
You can make a rectangle as long as you don't use all of the pieces (maybe that is part of the puzzle?). For example, from <a href="http://www.amazon.co.uk/Lychee-Tetris-Constructible-Three-dimensional-squares/dp/B00JZGD930" rel="nofollow">http://www.amazon.co.uk/Lychee-Tetris-Constructible-Three-di...</a>, remove the purple piece, shift the red piece to the left one space and flip it, then place the blue piece vertically on the right hand side.
I recently read a similar case in Simon Singh's book <i>Fermat's Enigma</i> <a href="http://www.amazon.com/reader/0385493622?_encoding=UTF8&query=15" rel="nofollow">http://www.amazon.com/reader/0385493622?_encoding=UTF8&query...</a> about the 14-15 puzzle. It was similarly unsolvable and provable the same way.<p>Interestingly, his account is rather different than that on wikipedia <a href="http://en.wikipedia.org/wiki/15_puzzle" rel="nofollow">http://en.wikipedia.org/wiki/15_puzzle</a>
Singh claims that Sam Lloyd created the puzzle, secretly proved it was impossible, and offered rewards to anyone who could solve it.
A similar trick can be used to solve the following problem: considering a rectangular grid, is it possible to find a cycle that goes from adjacent squares to adjacent squares and visits each square exactly once?<p>Answer (ROT13):
Vg qrcraqf ba gur cnevgl bs gur ahzore bs fdhnerf, juvpu qrcraqf ba jurgure gur qvzrafvbaf bs gur tevq fvqrf ner obgu bqq be abg. Vs gur ahzore bs fdhnerf vf bqq, gurer nera'g nf znal juvgr fdhnerf nf oynpx fdhnerf, naq n plpyr zhfg tb sebz juvgr gb oynpx naq sebz oynpx gb juvgr, fb ab plpyr rkvfgf. Vs vg vf rira, lbh pna rnfvyl pbzr hc jvgu n trareny fpurzr gb pbafgehpg n plpyr.
Speaking of Tetris proofs, I've had the following problem on the back-burner for a while: is there a way to check whether an arbitrary contiguous space comprised of squares can be filled in by tetrominoes? I don't have a math background so reasoning about it is difficult. Here's the question on StackOverflow: <a href="http://stackoverflow.com/questions/20083552/tetromino-space-filling-need-to-check-if-its-possible" rel="nofollow">http://stackoverflow.com/questions/20083552/tetromino-space-...</a>
Author here - slightly overwhelmed by the response, but thanks for everyone's feedback! I never thought so many people would be as interested as I am in the lamp...
Interesting concept. However, is it really a proof? If I removed piece #7 I'd have 24 boxes (12 white, 12 black) with which I could theoretically build a 6x4 rectangle, but I don't see how this would be actually possible with those pieces.<p>Maybe it only proves the negative, but not that there must be a solution.<p>Edit: there seems to be a solution for a 4x6 rectangle.
Very relevant to the video game Sigils of Elohim
<a href="http://store.steampowered.com/app/321480/" rel="nofollow">http://store.steampowered.com/app/321480/</a>
Reminds me of Knuth's Dancing Links algorithm: <a href="http://arxiv.org/pdf/cs/0011047.pdf" rel="nofollow">http://arxiv.org/pdf/cs/0011047.pdf</a>
If you remove the T piece, is it possible to assemble the other six into a rectangle? I suspect not, but the checkerboard proof does not suffice for this.