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Similar to: https://en.wikipedia.org/wiki/Mutilated_chessboard_problem
The problem and its proof are mentioned in EWD 1036: On the cruelty of really teaching computing science. http://www.cs.utexas.edu/users/EWD/transcriptions/EWD10xx/EW... >
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.
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 http://www.amazon.co.uk/Lychee-Tetris-Constructible-Three-di... , 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.
The argument in the article was that it's the purple piece (the "T") that is the problem (or least that makes the proof trivial). Can you make a rectangle with a subset that includes the T?
Every even x even rectangle has equal dark and light squares.
Every even x odd rectangle has equal dark and light squares.
Every odd x odd rectangle has one more odd then even, or one more even then odd. (1x1 square is obvious, any odd x odd rectangle can be reduced to 1x1 square by removing only odd x even rectangle.)
Since the T piece creates a odd-even difference of 2, any number of other tetris pieces (including repeats) that only contains a single T piece can never form a rectangle with no holes.
A few things: 1) Why limit yourself to 4x7? The 1988 NES version of Tetris is 10 units wide. 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). 3) In Tetris, a full row is removed immediately so having a complete rectangular shape that occupies the full available width is unrealistic. Pedantry aside, you'd have…
> 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. No devilry, there's just only so many ways you can connect four squares together to make a shape. http://en.wikipedia.org/wiki/Tetromino
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? 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 fd…
Earlier quoted context omitted.
If you buy 2 lamps, you've got 2 of all the 7 pieces. Piece 7 (the T-shaped piece) is doubled and you can switch the pattern for the second set. You'll have 15x color A and 13x color B in the first set and 13x color A and 15x color B in the second set. I'm pretty sure this would enable you to create a rectangular lamp.
Having the same number of both colours is not sufficient. 14 Z pieces will give you 28 of each color, but you can't make a rectangular lamp (of any size) from only Z pieces.
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: http://www.thinkgeek.com/product/f034/ (I realize we all have access to Google, but if I save 100 people 10 seconds, then I just saved 1000 seconds!)
The article links to it in the first sentence also: http://www.amazon.co.uk/Lychee-Tetris-Constructible-Three-di...
Earlier quoted context omitted.
Is that really obvious? I think the checkerboard pattern only says something about the neighborhoods of the pieces and the amounts of tiles of each color used, not whether there exists is a rectangle configuration.
If you buy 2 lamps, you've got 2 of all the 7 pieces. Piece 7 (the T-shaped piece) is doubled and you can switch the pattern for the second set. You'll have 15x color A and 13x color B in the first set and 13x color A and 15x color B in the second set. I'm pretty sure this would enable you to create a rectangular lamp.