Prime 15 Report post Posted February 7, 2013 Prove that you cannot cover a 10 x 10 chessboard with 25 figures (Problem from Russian Math Olympiads. 6-th grade, 1964.) Quote Share this post Link to post Share on other sites
0 bushindo 14 Report post Posted February 7, 2013 Prove that you cannot cover a 10 x 10 chessboard with 25 figuresTfig.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. 2 Quote Share this post Link to post Share on other sites
0 k-man 26 Report post Posted February 7, 2013 (edited) Prove that you cannot cover a 10 x 10 chessboard with 25 figuresTfig.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 February 7, 2013 by k-man Quote Share this post Link to post Share on other sites
0 Prime 15 Report post Posted February 7, 2013 (edited) 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 February 7, 2013 by Prime Quote Share this post Link to post Share on other sites
Prove that you cannot
cover a 10 x 10 chessboard with 25 figures
(Problem from Russian Math Olympiads. 6-th grade, 1964.)
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