Simple proof for tetris lamp
This lamp was given to me for my birthday last year. Wonderful little thing - you can move individual fragments, creating any shape, and they glow individually, feeding through conductive edges around the perimeter.
Because of the obvious connection with Tetris, one thing always annoyed me: it is impossible to make a lamp into a clean rectangle. No matter how hard I tried, there was always some piece sticking out from the side, and one was missing from the top, or another annoying combination was obtained.
This irritation spread to many who visited my room. In particular, one comrade spent the whole evening turning over fragments in different combinations and refusing to admit that someone had such a perverted mind and he successfully designed fragments that it is impossible to put together.
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His efforts were in vain. Since then, I have come to terms that it is probably impossible to make a lamp into a rectangle due to a specially selected set of fragments.
However, drinking last night in the room, another friend of mine (who had not been exposed to the immoral influence of the lamp before) saw the construction on the table, thought for a few minutes and came up with evidence that it really cannot be made into a rectangle. The proof was so simple and elegant that I decided to publish it here.
The lamp itself consists of seven separate parts: a total of 28 square fragments. Thus, if we want to form a regular figure, it should be 7x4 or 14x2. Here we show the first option simply because it has a more natural form. But the proof is valid for the second figure. Now imagine that we marked each square with a color — black or white — so that together they form a surface like a chessboard, as shown above. Note that the number of black cells must be equal to the number of white cells. It is this property that we will operate on.
So, it turns out 14 black cells and 14 white. If you look at each block separately, the problem immediately becomes obvious.
As you can see, for blocks 1-6, the number of black fragments is equal to the number of white. Naturally, the location of the white and black fragments depends on the position of the block in the rectangle, but the form itself indicates the number of such fragments (since the neighboring fragments must be of different colors).
However, block 7 violates harmony. Regardless of how to place it, it still consists of three fragments of the same color and one fragment of a different color, this property directly follows from its shape.
Thus, if you count the colors on all the blocks, you get 13 cells of the same color and 15 cells of a different color, regardless of the location of the blocks in the overall structure. But we need 14 fragments of each color, but we can’t get them, so the initial condition cannot be met, as required.
The proof itself is so simple: I am even disappointed that I myself have not found it before. Nevertheless, it is nice to know that you will not have to waste time on the mindless shuffling of fragments in the hope of a breakthrough.
Maybe I should transfer my annoyance from the lamp itself to the one who deliberately designed it in this way.
Because of the obvious connection with Tetris, one thing always annoyed me: it is impossible to make a lamp into a clean rectangle. No matter how hard I tried, there was always some piece sticking out from the side, and one was missing from the top, or another annoying combination was obtained.
This irritation spread to many who visited my room. In particular, one comrade spent the whole evening turning over fragments in different combinations and refusing to admit that someone had such a perverted mind and he successfully designed fragments that it is impossible to put together.
')
His efforts were in vain. Since then, I have come to terms that it is probably impossible to make a lamp into a rectangle due to a specially selected set of fragments.
However, drinking last night in the room, another friend of mine (who had not been exposed to the immoral influence of the lamp before) saw the construction on the table, thought for a few minutes and came up with evidence that it really cannot be made into a rectangle. The proof was so simple and elegant that I decided to publish it here.
The lamp itself consists of seven separate parts: a total of 28 square fragments. Thus, if we want to form a regular figure, it should be 7x4 or 14x2. Here we show the first option simply because it has a more natural form. But the proof is valid for the second figure. Now imagine that we marked each square with a color — black or white — so that together they form a surface like a chessboard, as shown above. Note that the number of black cells must be equal to the number of white cells. It is this property that we will operate on.
So, it turns out 14 black cells and 14 white. If you look at each block separately, the problem immediately becomes obvious.
As you can see, for blocks 1-6, the number of black fragments is equal to the number of white. Naturally, the location of the white and black fragments depends on the position of the block in the rectangle, but the form itself indicates the number of such fragments (since the neighboring fragments must be of different colors).
However, block 7 violates harmony. Regardless of how to place it, it still consists of three fragments of the same color and one fragment of a different color, this property directly follows from its shape.
Thus, if you count the colors on all the blocks, you get 13 cells of the same color and 15 cells of a different color, regardless of the location of the blocks in the overall structure. But we need 14 fragments of each color, but we can’t get them, so the initial condition cannot be met, as required.
Conclusion
The proof itself is so simple: I am even disappointed that I myself have not found it before. Nevertheless, it is nice to know that you will not have to waste time on the mindless shuffling of fragments in the hope of a breakthrough.
Maybe I should transfer my annoyance from the lamp itself to the one who deliberately designed it in this way.
Source: https://habr.com/ru/post/365977/
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