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The Simple Proof of the Tetris Lamp

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Re: The Simple Proof of the Tetris Lamp

#61

Post by ecesena »

rndn wrote: post #8
davecap1 wrote: post #2

That's exactly how they get you to buy TWO lamps! :)

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.

Here's an example. The trick I used was creating the extra block and the hole on the same side (so, not like the pic). With the pieces numbered as in the post:

    2233ffff
    72233ddd
    7744eaad
    7114eaag
    5114eegg
    555bbccg
    6666bbcc
Note that, as in the post, I'm not "respecting" Tetris in 2d: pieces 2 and 3 have been "3d rotated" and are "looking in the same direction".

Re: The Simple Proof of the Tetris Lamp

#62

Post by maaku »

AlexMax wrote: post #22
pharke wrote: post #10

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…

> 1) Why limit yourself to 4x7? The 1988 NES version of Tetris is 10 units wide. Presumably, the objective is to arrange the tetriminos into a pleasing rectangle shape. Unfortunately, the factors of 28 are 1, 2, 4, 7, 14 and 28, so 4x7 is the closest you can get to a square, and 2x14 has a similar problem of having the same number of 'white' squares as 'black' squares. The closest one can get is 5x6 with two holes re…

Why? A full row in tetris is removed. To someone who knows the game, leaving a hole in each row would be more pleasing. Aesthetically I'd also find it more interesting, but that's an aside.

Re: The Simple Proof of the Tetris Lamp

#64

Post by evanb »

nemetroid wrote: post #26
rmetzler wrote: post #25

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.

That's true, but here's a construction:

    OOZZ
    OOSZZ
    LLSS
    ILTS
    ILTT
    IJT 
    IJJJ
That's built out of one full set of OZSLJTI. If you take two of those, one rotated by 180˚, you get a rectangle.

Re: The Simple Proof of the Tetris Lamp

#65

Post by evanb »

dkersten wrote: post #58
nemetroid wrote: post #26

Earlier quoted context omitted.

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.

So the obvious next question is: how many sets do you need to create a perfect rectangle?

Two! (see my same-level comment)

Re: The Simple Proof of the Tetris Lamp

#66

Post by ecesena »

evanb wrote: post #65
dkersten wrote: post #58

Earlier quoted context omitted.

So the obvious next question is: how many sets do you need to create a perfect rectangle?

Two! (see my same-level comment)

So the next questions are:

- how many 8x7 solutions exist?

- does any 4x14 solution exist?

- does any 2x4x7 solution exist?

:)

Re: The Simple Proof of the Tetris Lamp

#67

Post by evanb »

ecesena wrote: post #66
evanb wrote: post #65

Earlier quoted context omitted.

Two! (see my same-level comment)

So the next questions are: - how many 8x7 solutions exist? - does any 4x14 solution exist? - does any 2x4x7 solution exist? :)

Let's stick to accept-reject questions for now :)

Re: The Simple Proof of the Tetris Lamp

#69

Post by Russell91 »

runn1ng wrote: post #47
pmelendez wrote: post #28

>"Maybe now I can shift my irritation from the lamp itself to whoever designed it to possess such a property. " I wouldn't be mad at the designer. That person designed a lamp and proposed an impossible puzzle that at first glance looked plausible. It is the best way to troll people... ever!

http://en.wikipedia.org/wiki/File:15-puzzle-loyd.svg :)

"Some later interest was fuelled by Loyd offering a $1,000 prize for anyone who could provide a solution for achieving a particular combination specified by Loyd, namely reversing the 14 and 15.[13] This was impossible, as had been shown over a decade earlier by Johnson & Story (1879), as it required a transformation from an even to an odd combination."

Re: The Simple Proof of the Tetris Lamp

#70

Post by pja »

gus_massa wrote: post #20
mjs wrote: post #6

Similar to: https://en.wikipedia.org/wiki/Mutilated_chessboard_problem

Yes. Every time you get a problem with a rectangle table, you must paint it like a chessboard. If that is not enough to find the solution, you must try to think. A small technical detail, dew to professional deformation. The article says that the numbers of squares of each color must be the same, but that only happens if the total number is even, like in this case 13+15=28. If the total number of squares is odd, then…

Minor derail - I know I mistype 'their' as 'they're' and likewise for other similar homonym pairs all the time, but due -> dew is a particularly nice example of the genre :)
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