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Guest Message by DevFuse

Origins of Tetris?

Started by Prime, Feb 07 2013 09:03 PM

Best Answer bushindo, 07 February 2013 - 10:49 PM

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

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

Answer

Spoiler for

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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3 replies to this topic

#1 Prime

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Posted 07 February 2013 - 09:03 PM

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

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

0

Past prime, actually.

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#2 bushindo

Senior Member

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717 posts

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Location:Los Angeles, CA

Posted 07 February 2013 - 10:49 PM Best Answer

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

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

Answer

Spoiler for

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.

2

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#3 k-man

Advanced Member

Members
361 posts

Gender:Male

Posted 07 February 2013 - 10:58 PM

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

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

Answer
Spoiler for
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, 07 February 2013 - 11:04 PM.

0

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#4 Prime

Senior Member

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838 posts

Gender:Male
Location:Illinois, US

Posted 07 February 2013 - 11:23 PM

Answer
Spoiler for
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.

Spoiler for or you could put it like that:

I knew, this problem would not last long here.

Edited by Prime, 07 February 2013 - 11:32 PM.

0

Past prime, actually.

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Origins of Tetris?

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#4 Prime

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