All neighbor sum is 0 on a chessboard
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
asked 3 hours ago
Oray
15.6k435149
add a comment |
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
asked 3 hours ago
Oray
15.6k435149
add a comment |
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
asked 3 hours ago
Oray
15.6k435149
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
asked 3 hours ago
Oray
15.6k435149
asked 3 hours ago
Oray
15.6k435149
edited 3 hours ago
asked 3 hours ago
Oray
15.6k435149
asked 3 hours ago
Oray
15.6k435149
asked 3 hours ago
Oray
15.6k435149
15.6k435149
add a comment |
add a comment |
2 Answers
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oldest
votes
Least amount of zero valued cells is
zero
One such possible grid is:
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?
answered 2 hours ago
Mohit Jain
2,8351841
add a comment |
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.
answered 31 mins ago
JonMark Perry
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
11 mins ago
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
Least amount of zero valued cells is
zero
One such possible grid is:
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?
answered 2 hours ago
Mohit Jain
2,8351841
add a comment |
Least amount of zero valued cells is
zero
One such possible grid is:
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?
answered 2 hours ago
Mohit Jain
2,8351841
add a comment |
Least amount of zero valued cells is
zero
One such possible grid is:
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?
answered 2 hours ago
Mohit Jain
2,8351841
Least amount of zero valued cells is
zero
One such possible grid is:
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?
answered 2 hours ago
Mohit Jain
2,8351841
edited 2 hours ago
answered 2 hours ago
Mohit Jain
2,8351841
answered 2 hours ago
Mohit Jain
2,8351841
answered 2 hours ago
Mohit Jain
2,8351841
2,8351841
add a comment |
add a comment |
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.
answered 31 mins ago
JonMark Perry
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
11 mins ago
add a comment |
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.
answered 31 mins ago
JonMark Perry
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
11 mins ago
add a comment |
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.
answered 31 mins ago
JonMark Perry
17.7k63585
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.
answered 31 mins ago
JonMark Perry
17.7k63585
answered 31 mins ago
JonMark Perry
17.7k63585
answered 31 mins ago
JonMark Perry
17.7k63585
answered 31 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
11 mins ago
add a comment |
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
11 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
11 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
11 mins ago
add a comment |
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