Prime Posted February 7, 2013 Report Share 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 Link to comment Share on other sites More sharing options...
0 bushindo Posted February 7, 2013 Report Share 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 Link to comment Share on other sites More sharing options...
0 k-man Posted February 7, 2013 Report Share 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 Link to comment Share on other sites More sharing options...
0 Prime Posted February 7, 2013 Author Report Share 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 Link to comment Share on other sites More sharing options...
Question
Prime
Prove that you cannot
cover a 10 x 10 chessboard with 25 figures
(Problem from Russian Math Olympiads. 6-th grade, 1964.)
Link to comment
Share on other sites
3 answers to this question
Recommended Posts
Join the conversation
You can post now and register later. If you have an account, sign in now to post with your account.