All neighbor sum is 0 on a chessboard












8














Neighbor cells of a cell is defined as all the cells where they have common edge of a cell in a matrix. A matrix is formed by putting numerical values in each cell with real numbers.



Form a matrix with the dimension of $8$x$8$ (like a chessboard) where sum of all neighbors of each cell will be $0$.




What is the least amount of 0 valued cells possible after forming the matrix?







mathematics logical-deduction







share|improve this question





























    8














    Neighbor cells of a cell is defined as all the cells where they have common edge of a cell in a matrix. A matrix is formed by putting numerical values in each cell with real numbers.



    Form a matrix with the dimension of $8$x$8$ (like a chessboard) where sum of all neighbors of each cell will be $0$.




    What is the least amount of 0 valued cells possible after forming the matrix?







    mathematics logical-deduction







    share|improve this question



























      8












      8








      8


      1





      Neighbor cells of a cell is defined as all the cells where they have common edge of a cell in a matrix. A matrix is formed by putting numerical values in each cell with real numbers.



      Form a matrix with the dimension of $8$x$8$ (like a chessboard) where sum of all neighbors of each cell will be $0$.




      What is the least amount of 0 valued cells possible after forming the matrix?







      mathematics logical-deduction







      share|improve this question















      Neighbor cells of a cell is defined as all the cells where they have common edge of a cell in a matrix. A matrix is formed by putting numerical values in each cell with real numbers.



      Form a matrix with the dimension of $8$x$8$ (like a chessboard) where sum of all neighbors of each cell will be $0$.




      What is the least amount of 0 valued cells possible after forming the matrix?







      mathematics logical-deduction




      mathematics logical-deduction






      share|improve this question















      share|improve this question













      share|improve this question




      share|improve this question








      edited yesterday







      Oray

















      asked yesterday









      OrayOray

      15.6k435150




      15.6k435150






















          2 Answers
          2






          active

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          12














          This works for any given $a,b,c,d$:




          $$begin{array} {|c|c|c|c|c|c|c|c|}hline a&b&c&d&d&c&b&a\hline -b&-a-c&-b-d&-c-d&-c-d&-b-d&-a-c&-b\hline c&b+d&a+c+d&b+c+d&b+c+d&a+c+d&b+d&c\hline -d&-c-d&-b-c-d&-a-b-c-d&-a-b-c-d&-b-c-d&-c-d&-d\hline d&c+d&b+c+d&a+b+c+d&a+b+c+d&b+c+d&c+d&d\hline -c&-b-d&-a-c-d&-b-c-d&-b-c-d&-a-c-d&-b-d&-c\hline b&a+c&b+d&c+d&c+d&b+d&a+c&b\hline -a&-b&-c&-d&-d&-c&-b&-a\hline end{array}$$




          so:




          zero zeroes are needed (for example, $a,b,c,dgt0$)




          Following @Jaap's comment:




          $$begin{array} {|c|c|c|c|c|c|c|c|}hline a&h&c&g&d&f&b&e\hline -h&-a-c&-h-g&-c-d&-f-g&-b-d&-e-f&-b\hline c&g+h&a+c+d&f+g+h&b+c+d&e+f+h&b+d&f\hline -g&-c-d&-f-g-h&-a-b-c-d&-e-f-g-h&-b-c-d&-f-h&-d\hline d&f+g&b+c+d&e+f+g+h&a+b+c+d&f+g+h&c+d&g\hline -f&-b-d&-e-f-g&-b-c-d&-f-g-h&-a-c-d&-g-h&-c\hline b&e+f&b+d&f+h&c+d&g+h&a+c&h\hline -e&-b&-f&-d&-g&-c&-h&-a\hline end{array}$$







          share|improve this answer



















          • 6




            You could consider the white squares of the chessboard separately from the black squares, treating them as independent, identical puzzles. You have given those two sets the same (mirrored) solution, but you can choose different $a,b,c,d$ for them. This gives you the most generic parametric solution, with 8 variables on the first row of the board.
            – Jaap Scherphuis
            yesterday



















          5














          Least amount of zero valued cells is




          zero




          One such possible grid is:




          enter image description here




          Because the grid size is even in both directions, I used the following algorithms to fill it:




          1. Fill 1 (or -1) in first two cells. Some other number will also work.

          2. Copy the same number to the second end of row and zero sum inverse in opposite col (or vice versa).

          3. Fill rest of the number in a way that constraints are met.

          4. If any of the cells is zero, try with different numbers.




          I got the above algorithm as an intuition. I am still figuring an easy way to explain/prove why does it work?






          share|improve this answer























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            active

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            12














            This works for any given $a,b,c,d$:




            $$begin{array} {|c|c|c|c|c|c|c|c|}hline a&b&c&d&d&c&b&a\hline -b&-a-c&-b-d&-c-d&-c-d&-b-d&-a-c&-b\hline c&b+d&a+c+d&b+c+d&b+c+d&a+c+d&b+d&c\hline -d&-c-d&-b-c-d&-a-b-c-d&-a-b-c-d&-b-c-d&-c-d&-d\hline d&c+d&b+c+d&a+b+c+d&a+b+c+d&b+c+d&c+d&d\hline -c&-b-d&-a-c-d&-b-c-d&-b-c-d&-a-c-d&-b-d&-c\hline b&a+c&b+d&c+d&c+d&b+d&a+c&b\hline -a&-b&-c&-d&-d&-c&-b&-a\hline end{array}$$




            so:




            zero zeroes are needed (for example, $a,b,c,dgt0$)




            Following @Jaap's comment:




            $$begin{array} {|c|c|c|c|c|c|c|c|}hline a&h&c&g&d&f&b&e\hline -h&-a-c&-h-g&-c-d&-f-g&-b-d&-e-f&-b\hline c&g+h&a+c+d&f+g+h&b+c+d&e+f+h&b+d&f\hline -g&-c-d&-f-g-h&-a-b-c-d&-e-f-g-h&-b-c-d&-f-h&-d\hline d&f+g&b+c+d&e+f+g+h&a+b+c+d&f+g+h&c+d&g\hline -f&-b-d&-e-f-g&-b-c-d&-f-g-h&-a-c-d&-g-h&-c\hline b&e+f&b+d&f+h&c+d&g+h&a+c&h\hline -e&-b&-f&-d&-g&-c&-h&-a\hline end{array}$$







            share|improve this answer



















            • 6




              You could consider the white squares of the chessboard separately from the black squares, treating them as independent, identical puzzles. You have given those two sets the same (mirrored) solution, but you can choose different $a,b,c,d$ for them. This gives you the most generic parametric solution, with 8 variables on the first row of the board.
              – Jaap Scherphuis
              yesterday
















            12














            This works for any given $a,b,c,d$:




            $$begin{array} {|c|c|c|c|c|c|c|c|}hline a&b&c&d&d&c&b&a\hline -b&-a-c&-b-d&-c-d&-c-d&-b-d&-a-c&-b\hline c&b+d&a+c+d&b+c+d&b+c+d&a+c+d&b+d&c\hline -d&-c-d&-b-c-d&-a-b-c-d&-a-b-c-d&-b-c-d&-c-d&-d\hline d&c+d&b+c+d&a+b+c+d&a+b+c+d&b+c+d&c+d&d\hline -c&-b-d&-a-c-d&-b-c-d&-b-c-d&-a-c-d&-b-d&-c\hline b&a+c&b+d&c+d&c+d&b+d&a+c&b\hline -a&-b&-c&-d&-d&-c&-b&-a\hline end{array}$$




            so:




            zero zeroes are needed (for example, $a,b,c,dgt0$)




            Following @Jaap's comment:




            $$begin{array} {|c|c|c|c|c|c|c|c|}hline a&h&c&g&d&f&b&e\hline -h&-a-c&-h-g&-c-d&-f-g&-b-d&-e-f&-b\hline c&g+h&a+c+d&f+g+h&b+c+d&e+f+h&b+d&f\hline -g&-c-d&-f-g-h&-a-b-c-d&-e-f-g-h&-b-c-d&-f-h&-d\hline d&f+g&b+c+d&e+f+g+h&a+b+c+d&f+g+h&c+d&g\hline -f&-b-d&-e-f-g&-b-c-d&-f-g-h&-a-c-d&-g-h&-c\hline b&e+f&b+d&f+h&c+d&g+h&a+c&h\hline -e&-b&-f&-d&-g&-c&-h&-a\hline end{array}$$







            share|improve this answer



















            • 6




              You could consider the white squares of the chessboard separately from the black squares, treating them as independent, identical puzzles. You have given those two sets the same (mirrored) solution, but you can choose different $a,b,c,d$ for them. This gives you the most generic parametric solution, with 8 variables on the first row of the board.
              – Jaap Scherphuis
              yesterday














            12












            12








            12






            This works for any given $a,b,c,d$:




            $$begin{array} {|c|c|c|c|c|c|c|c|}hline a&b&c&d&d&c&b&a\hline -b&-a-c&-b-d&-c-d&-c-d&-b-d&-a-c&-b\hline c&b+d&a+c+d&b+c+d&b+c+d&a+c+d&b+d&c\hline -d&-c-d&-b-c-d&-a-b-c-d&-a-b-c-d&-b-c-d&-c-d&-d\hline d&c+d&b+c+d&a+b+c+d&a+b+c+d&b+c+d&c+d&d\hline -c&-b-d&-a-c-d&-b-c-d&-b-c-d&-a-c-d&-b-d&-c\hline b&a+c&b+d&c+d&c+d&b+d&a+c&b\hline -a&-b&-c&-d&-d&-c&-b&-a\hline end{array}$$




            so:




            zero zeroes are needed (for example, $a,b,c,dgt0$)




            Following @Jaap's comment:




            $$begin{array} {|c|c|c|c|c|c|c|c|}hline a&h&c&g&d&f&b&e\hline -h&-a-c&-h-g&-c-d&-f-g&-b-d&-e-f&-b\hline c&g+h&a+c+d&f+g+h&b+c+d&e+f+h&b+d&f\hline -g&-c-d&-f-g-h&-a-b-c-d&-e-f-g-h&-b-c-d&-f-h&-d\hline d&f+g&b+c+d&e+f+g+h&a+b+c+d&f+g+h&c+d&g\hline -f&-b-d&-e-f-g&-b-c-d&-f-g-h&-a-c-d&-g-h&-c\hline b&e+f&b+d&f+h&c+d&g+h&a+c&h\hline -e&-b&-f&-d&-g&-c&-h&-a\hline end{array}$$







            share|improve this answer














            This works for any given $a,b,c,d$:




            $$begin{array} {|c|c|c|c|c|c|c|c|}hline a&b&c&d&d&c&b&a\hline -b&-a-c&-b-d&-c-d&-c-d&-b-d&-a-c&-b\hline c&b+d&a+c+d&b+c+d&b+c+d&a+c+d&b+d&c\hline -d&-c-d&-b-c-d&-a-b-c-d&-a-b-c-d&-b-c-d&-c-d&-d\hline d&c+d&b+c+d&a+b+c+d&a+b+c+d&b+c+d&c+d&d\hline -c&-b-d&-a-c-d&-b-c-d&-b-c-d&-a-c-d&-b-d&-c\hline b&a+c&b+d&c+d&c+d&b+d&a+c&b\hline -a&-b&-c&-d&-d&-c&-b&-a\hline end{array}$$




            so:




            zero zeroes are needed (for example, $a,b,c,dgt0$)




            Following @Jaap's comment:




            $$begin{array} {|c|c|c|c|c|c|c|c|}hline a&h&c&g&d&f&b&e\hline -h&-a-c&-h-g&-c-d&-f-g&-b-d&-e-f&-b\hline c&g+h&a+c+d&f+g+h&b+c+d&e+f+h&b+d&f\hline -g&-c-d&-f-g-h&-a-b-c-d&-e-f-g-h&-b-c-d&-f-h&-d\hline d&f+g&b+c+d&e+f+g+h&a+b+c+d&f+g+h&c+d&g\hline -f&-b-d&-e-f-g&-b-c-d&-f-g-h&-a-c-d&-g-h&-c\hline b&e+f&b+d&f+h&c+d&g+h&a+c&h\hline -e&-b&-f&-d&-g&-c&-h&-a\hline end{array}$$








            share|improve this answer














            share|improve this answer



            share|improve this answer








            edited yesterday

























            answered yesterday









            JonMark PerryJonMark Perry

            17.8k63686




            17.8k63686








            • 6




              You could consider the white squares of the chessboard separately from the black squares, treating them as independent, identical puzzles. You have given those two sets the same (mirrored) solution, but you can choose different $a,b,c,d$ for them. This gives you the most generic parametric solution, with 8 variables on the first row of the board.
              – Jaap Scherphuis
              yesterday














            • 6




              You could consider the white squares of the chessboard separately from the black squares, treating them as independent, identical puzzles. You have given those two sets the same (mirrored) solution, but you can choose different $a,b,c,d$ for them. This gives you the most generic parametric solution, with 8 variables on the first row of the board.
              – Jaap Scherphuis
              yesterday








            6




            6




            You could consider the white squares of the chessboard separately from the black squares, treating them as independent, identical puzzles. You have given those two sets the same (mirrored) solution, but you can choose different $a,b,c,d$ for them. This gives you the most generic parametric solution, with 8 variables on the first row of the board.
            – Jaap Scherphuis
            yesterday




            You could consider the white squares of the chessboard separately from the black squares, treating them as independent, identical puzzles. You have given those two sets the same (mirrored) solution, but you can choose different $a,b,c,d$ for them. This gives you the most generic parametric solution, with 8 variables on the first row of the board.
            – Jaap Scherphuis
            yesterday











            5














            Least amount of zero valued cells is




            zero




            One such possible grid is:




            enter image description here




            Because the grid size is even in both directions, I used the following algorithms to fill it:




            1. Fill 1 (or -1) in first two cells. Some other number will also work.

            2. Copy the same number to the second end of row and zero sum inverse in opposite col (or vice versa).

            3. Fill rest of the number in a way that constraints are met.

            4. If any of the cells is zero, try with different numbers.




            I got the above algorithm as an intuition. I am still figuring an easy way to explain/prove why does it work?






            share|improve this answer




























              5














              Least amount of zero valued cells is




              zero




              One such possible grid is:




              enter image description here




              Because the grid size is even in both directions, I used the following algorithms to fill it:




              1. Fill 1 (or -1) in first two cells. Some other number will also work.

              2. Copy the same number to the second end of row and zero sum inverse in opposite col (or vice versa).

              3. Fill rest of the number in a way that constraints are met.

              4. If any of the cells is zero, try with different numbers.




              I got the above algorithm as an intuition. I am still figuring an easy way to explain/prove why does it work?






              share|improve this answer


























                5












                5








                5






                Least amount of zero valued cells is




                zero




                One such possible grid is:




                enter image description here




                Because the grid size is even in both directions, I used the following algorithms to fill it:




                1. Fill 1 (or -1) in first two cells. Some other number will also work.

                2. Copy the same number to the second end of row and zero sum inverse in opposite col (or vice versa).

                3. Fill rest of the number in a way that constraints are met.

                4. If any of the cells is zero, try with different numbers.




                I got the above algorithm as an intuition. I am still figuring an easy way to explain/prove why does it work?






                share|improve this answer














                Least amount of zero valued cells is




                zero




                One such possible grid is:




                enter image description here




                Because the grid size is even in both directions, I used the following algorithms to fill it:




                1. Fill 1 (or -1) in first two cells. Some other number will also work.

                2. Copy the same number to the second end of row and zero sum inverse in opposite col (or vice versa).

                3. Fill rest of the number in a way that constraints are met.

                4. If any of the cells is zero, try with different numbers.




                I got the above algorithm as an intuition. I am still figuring an easy way to explain/prove why does it work?







                share|improve this answer














                share|improve this answer



                share|improve this answer








                edited yesterday

























                answered yesterday









                Mohit JainMohit Jain

                2,8551841




                2,8551841






























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