All neighbor sum is 0 on a chessboard












4














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





























    4














    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



























      4












      4








      4







      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 3 hours ago

























      asked 3 hours ago









      Oray

      15.6k435149




      15.6k435149






















          2 Answers
          2






          active

          oldest

          votes


















          2














          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.




          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































            2














            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.







            share|improve this answer





















            • very good explanation and a proof, but not any a,b,c,d actually, but any a,b,c,d where their any kind of sum or itself is not 0 :)
              – Oray
              2 mins ago













            Your Answer





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            2 Answers
            2






            active

            oldest

            votes








            2 Answers
            2






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            2














            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.




            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




























              2














              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.




              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


























                2












                2








                2






                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.




                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.




                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 1 hour ago

























                answered 2 hours ago









                Mohit Jain

                2,8251841




                2,8251841























                    2














                    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.







                    share|improve this answer





















                    • very good explanation and a proof, but not any a,b,c,d actually, but any a,b,c,d where their any kind of sum or itself is not 0 :)
                      – Oray
                      2 mins ago


















                    2














                    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.







                    share|improve this answer





















                    • very good explanation and a proof, but not any a,b,c,d actually, but any a,b,c,d where their any kind of sum or itself is not 0 :)
                      – Oray
                      2 mins ago
















                    2












                    2








                    2






                    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.







                    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.








                    share|improve this answer












                    share|improve this answer



                    share|improve this answer










                    answered 23 mins ago









                    JonMark Perry

                    17.7k63585




                    17.7k63585












                    • very good explanation and a proof, but not any a,b,c,d actually, but any a,b,c,d where their any kind of sum or itself is not 0 :)
                      – Oray
                      2 mins ago




















                    • very good explanation and a proof, but not any a,b,c,d actually, but any a,b,c,d where their any kind of sum or itself is not 0 :)
                      – Oray
                      2 mins ago


















                    very good explanation and a proof, but not any a,b,c,d actually, but any a,b,c,d where their any kind of sum or itself is not 0 :)
                    – Oray
                    2 mins ago






                    very good explanation and a proof, but not any a,b,c,d actually, but any a,b,c,d where their any kind of sum or itself is not 0 :)
                    – Oray
                    2 mins ago




















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