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Prime

Origins of Tetris?

Question

Prove that you cannot
cover a 10 x 10 chessboard with 25 figures

post-9379-0-41430000-1360267246_thumb.gi


(Problem from Russian Math Olympiads. 6-th grade, 1964.)

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Prove that you cannot

cover a 10 x 10 chessboard with 25 figures

attachicon.gifTfig.gif

(Problem from Russian Math Olympiads. 6-th grade, 1964.)

Answer

Let's say that the 10x10 grid is a chessboard. There would then be 50 Black and 50 White cells. Each tetris piece would be one of two types

(1)
B
W B
B

(2)
W
B W
W

So, let A be the number of pieces of type (1), and B be the number of pieces of type (2). The following two equations would have to be true if we can fill a 10x10 board

A + B = 25

3*A + B = 50

But obviously, there are no integer solutions for the above, so we can't fill the board with this Tetris shape.

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Prove that you cannot

cover a 10 x 10 chessboard with 25 figures

attachicon.gifTfig.gif

(Problem from Russian Math Olympiads. 6-th grade, 1964.)

Answer

Let's say that the 10x10 grid is a chessboard. There would then be 50 Black and 50 White cells. Each tetris piece would be one of two types

(1)
B
W B
B

(2)
W
B W
W

So, let A be the number of pieces of type (1), and B be the number of pieces of type (2). The following two equations would have to be true if we can fill a 10x10 board

A + B = 25

3*A + B = 50

But obviously, there are no integer solutions for the above, so we can't fill the board with this Tetris shape.

Nice! I had a proof that involved reviewing different scenarios, but I'm not going to post it as it's not as elegant as this.

Edited by k-man

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Answer

Let's say that the 10x10 grid is a chessboard. There would then be 50 Black and 50 White cells. Each tetris piece would be one of two types

(1)
B
W B
B

(2)
W
B W
W

So, let A be the number of pieces of type (1), and B be the number of pieces of type (2). The following two equations would have to be true if we can fill a 10x10 board

A + B = 25

3*A + B = 50

But obviously, there are no integer solutions for the above, so we can't fill the board with this Tetris shape.

Yes, that's the solution.

You need an equal number of type A and type B figures, and 25 is not divisible by 2.

I knew, this problem would not last long here.

Edited by Prime

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