# Count of cells in a matrix whose adjacent cells’s sum is prime Number

Given a M x N matrix mat[][], the task is to count the number of cells which have the sum of its adjacent cells equal to a prime number. For a cell x[i][j], only x[i+1][j], x[i-1][j], x[i][j+1] and x[i][j-1] are the adjacent cells.
Examples:

Input : mat[][] = {{1, 3}, {2, 5}}
Output :
Explanation: Only the cells mat[0][0] and mat[1][1] satisfying the condition.
i.e for mat[0][0]:(3+2) = 5, for mat[1][1]: (3+2) = 5
Input : mat[][] = {{1, 2, 3, 4}, {5, 6, 7, 8}, {9, 10, 11, 12}}
Output :
Explanation: Cells mat[0][0], mat[0][2], mat[0][3], mat[1][3], mat[2][2] and mat[2][3] are satisfying the condition.

Prerequisites: Sieve of Eratosthenes
Approach:

• Precompute and store the prime numbers using Sieve.
• Iterate the entire matrix and for each cell find the sum of all adjacent cells.
• If the sum of adjacent cells equal to a prime number then increments the count.
• Return the value of the count.

Below is the implementation of the above approach.

 `// CPP program to find the cells whose` `// adjacent cells's sum is prime Number` `#include ` `using` `namespace` `std;` `#define MAX 100005`   `bool` `prime[MAX];`   `void` `SieveOfEratosthenes()` `{` `    ``// Create a boolean array "prime[0..MAX-1]"` `    ``// and initialize all entries it as true.` `    ``// A value in prime[i] will finally` `    ``// be false if i is Not a prime, else true.` `    ``memset``(prime, ``true``, ``sizeof``(prime));`   `    ``prime[0] = prime[1] = ``false``;`   `    ``for` `(``int` `p = 2; p * p < MAX; p++) {` `        ``// If prime[p] is not changed,` `        ``// then it is a prime` `        ``if` `(prime[p] == ``true``) {` `            ``// Update all multiples of p` `            ``// greater than or` `            ``// equal to the square of it` `            ``// numbers which are multiple of p and are` `            ``// less than p^2 are already been marked.` `            ``for` `(``int` `i = p * p; i < MAX; i += p)` `                ``prime[i] = ``false``;` `        ``}` `    ``}` `}`   `// Function to count the cells having` `// adjacent cell's sum` `// is equal to prime` `int` `PrimeSumCells(vector >& mat)` `{` `    ``int` `count = 0;`   `    ``int` `N = mat.size();` `    ``int` `M = mat[0].size();`   `    ``// Traverse for all the cells` `    ``for` `(``int` `i = 0; i < N; i++) {` `        ``for` `(``int` `j = 0; j < M; j++) {`   `            ``int` `sum = 0;`   `            ``// i-1, j` `            ``if` `(i - 1 >= 0)` `                ``sum += mat[i - 1][j];`   `            ``// i+1, j` `            ``if` `(i + 1 < N)` `                ``sum += mat[i + 1][j];`   `            ``// i, j-1` `            ``if` `(j - 1 >= 0)` `                ``sum += mat[i][j - 1];`   `            ``// i, j+1` `            ``if` `(j + 1 < M)` `                ``sum += mat[i][j + 1];`   `            ``// If the sum is a prime number` `            ``if` `(prime[sum])` `                ``count++;` `        ``}` `    ``}`   `    ``// Return the count` `    ``return` `count;` `}`   `// Driver Program` `int` `main()` `{` `    ``SieveOfEratosthenes();`   `    ``vector > mat = { { 1, 2, 3, 4 },` `                                 ``{ 5, 6, 7, 8 },` `                                 ``{ 9, 10, 11, 12 } };`   `    ``// Function call` `    ``cout << PrimeSumCells(mat) << endl;` `}`

 `// Java program to find the cells whose` `// adjacent cells's sum is prime Number` `class` `GFG{` `static` `final` `int` `MAX = ``100005``;`   `static` `boolean` `[]prime = ``new` `boolean``[MAX];`   `static` `void` `SieveOfEratosthenes()` `{` `    ``// Create a boolean array "prime[0..MAX-1]"` `    ``// and initialize all entries it as true.` `    ``// A value in prime[i] will finally` `    ``// be false if i is Not a prime, else true.` `    ``for` `(``int` `i = ``0``; i < prime.length; i++)` `    ``prime[i] = ``true``;`   `    ``prime[``0``] = prime[``1``] = ``false``;`   `    ``for` `(``int` `p = ``2``; p * p < MAX; p++)` `    ``{` `        ``// If prime[p] is not changed,` `        ``// then it is a prime` `        ``if` `(prime[p] == ``true``)` `        ``{` `            ``// Update all multiples of p` `            ``// greater than or` `            ``// equal to the square of it` `            ``// numbers which are multiple of p and are` `            ``// less than p^2 are already been marked.` `            ``for` `(``int` `i = p * p; i < MAX; i += p)` `                ``prime[i] = ``false``;` `        ``}` `    ``}` `}`   `// Function to count the cells having` `// adjacent cell's sum` `// is equal to prime` `static` `int` `PrimeSumCells(``int` `[][]mat)` `{` `    ``int` `count = ``0``;`   `    ``int` `N = mat.length;` `    ``int` `M = mat[``0``].length;`   `    ``// Traverse for all the cells` `    ``for` `(``int` `i = ``0``; i < N; i++) ` `    ``{` `        ``for` `(``int` `j = ``0``; j < M; j++)` `        ``{` `            ``int` `sum = ``0``;`   `            ``// i-1, j` `            ``if` `(i - ``1` `>= ``0``)` `                ``sum += mat[i - ``1``][j];`   `            ``// i+1, j` `            ``if` `(i + ``1` `< N)` `                ``sum += mat[i + ``1``][j];`   `            ``// i, j-1` `            ``if` `(j - ``1` `>= ``0``)` `                ``sum += mat[i][j - ``1``];`   `            ``// i, j+1` `            ``if` `(j + ``1` `< M)` `                ``sum += mat[i][j + ``1``];`   `            ``// If the sum is a prime number` `            ``if` `(prime[sum])` `                ``count++;` `        ``}` `    ``}`   `    ``// Return the count` `    ``return` `count;` `}`   `// Driver Code` `public` `static` `void` `main(String[] args)` `{` `    ``SieveOfEratosthenes();`   `    ``int` `[][]mat = { { ``1``, ``2``, ``3``, ``4` `},` `                    ``{ ``5``, ``6``, ``7``, ``8` `},` `                    ``{ ``9``, ``10``, ``11``, ``12` `} };`   `    ``// Function call` `    ``System.out.print(PrimeSumCells(mat) + ``"\n"``);` `}` `}`   `// This code is contributed by sapnasingh4991`

 `# Python 3 program to ` `# find the cells whose` `# adjacent cells's ` `# sum is prime Number` `MAX` `=` `100005` `prime ``=` `[``True``] ``*` `MAX`   `def` `SieveOfEratosthenes():`   `    ``# Create a boolean array "prime[0..MAX-1]"` `    ``# and initialize all entries it as true.` `    ``# A value in prime[i] will finally` `    ``# be false if i is Not a prime, else true.` `    ``global` `prime` `    `  `    ``prime[``0``] ``=` `prime[``1``] ``=` `False`   `    ``p ``=` `2` `    ``while` `p ``*` `p < ``MAX``:` `      `  `        ``# If prime[p] is not changed,` `        ``# then it is a prime` `        ``if` `(prime[p] ``=``=` `True``):` `          `  `            ``# Update all multiples of p` `            ``# greater than or` `            ``# equal to the square of it` `            ``# numbers which are multiple of ` `            ``# p and are less than p^2 are ` `            ``# already been marked.` `            ``for` `i ``in` `range` `(p ``*` `p, ``MAX``, p):` `                ``prime[i] ``=` `False`                `        ``p ``+``=` `1` `      `  `# Function to count the ` `# cells having adjacent ` `# cell's sum is equal to prime` `def` `PrimeSumCells(mat):`   `    ``count ``=` `0` `    ``N ``=` `len``(mat)` `    ``M ``=` `len``(mat[``0``])`   `    ``# Traverse for all the cells` `    ``for` `i ``in` `range` `(N):` `        ``for` `j ``in` `range` `(M):`   `            ``sum` `=` `0`   `            ``# i - 1, j` `            ``if` `(i ``-` `1` `>``=` `0``):` `                ``sum` `+``=` `mat[i ``-` `1``][j]`   `            ``# i + 1, j` `            ``if` `(i ``+` `1` `< N):` `                ``sum` `+``=` `mat[i ``+` `1``][j]`   `            ``# i, j - 1` `            ``if` `(j ``-` `1` `>``=` `0``):` `                ``sum` `+``=` `mat[i][j ``-` `1``]`   `            ``# i, j + 1` `            ``if` `(j ``+` `1` `< M):` `                ``sum` `+``=` `mat[i][j ``+` `1``]`   `            ``# If the sum is a prime number` `            ``if` `(prime[``sum``]):` `                ``count ``+``=` `1` `   `  `    ``# Return the count` `    ``return` `count`   `# Driver code` `if` `__name__ ``=``=``"__main__"``:` `      `  `    ``SieveOfEratosthenes()` `    ``mat ``=` `[[``1``, ``2``, ``3``, ``4``],` `           ``[``5``, ``6``, ``7``, ``8``],` `           ``[``9``, ``10``, ``11``, ``12``]]`   `    ``# Function call` `    ``print` `(PrimeSumCells(mat))` `    `  `# This code is contributed by Chitranayal`

 `// C# program to find the cells whose` `// adjacent cells's sum is prime Number` `using` `System;` `class` `GFG{` `    `  `static` `readonly` `int` `MAX = 100005;` `static` `bool` `[]prime = ``new` `bool``[MAX];`   `static` `void` `SieveOfEratosthenes()` `{` `    ``// Create a bool array "prime[0..MAX-1]"` `    ``// and initialize all entries it as true.` `    ``// A value in prime[i] will finally` `    ``// be false if i is Not a prime, else true.` `    ``for` `(``int` `i = 0; i < prime.Length; i++)` `    ``prime[i] = ``true``;`   `    ``prime[0] = prime[1] = ``false``;`   `    ``for` `(``int` `p = 2; p * p < MAX; p++)` `    ``{` `        ``// If prime[p] is not changed,` `        ``// then it is a prime` `        ``if` `(prime[p] == ``true``)` `        ``{` `            ``// Update all multiples of p` `            ``// greater than or` `            ``// equal to the square of it` `            ``// numbers which are multiple of p and are` `            ``// less than p^2 are already been marked.` `            ``for` `(``int` `i = p * p; i < MAX; i += p)` `                ``prime[i] = ``false``;` `        ``}` `    ``}` `}`   `// Function to count the cells having` `// adjacent cell's sum` `// is equal to prime` `static` `int` `PrimeSumCells(``int` `[,]mat)` `{` `    ``int` `count = 0;`   `    ``int` `N = mat.GetLength(0);` `    ``int` `M = mat.GetLength(1);`   `    ``// Traverse for all the cells` `    ``for` `(``int` `i = 0; i < N; i++) ` `    ``{` `        ``for` `(``int` `j = 0; j < M; j++)` `        ``{` `            ``int` `sum = 0;`   `            ``// i-1, j` `            ``if` `(i - 1 >= 0)` `                ``sum += mat[i - 1, j];`   `            ``// i+1, j` `            ``if` `(i + 1 < N)` `                ``sum += mat[i + 1, j];`   `            ``// i, j-1` `            ``if` `(j - 1 >= 0)` `                ``sum += mat[i, j - 1];`   `            ``// i, j+1` `            ``if` `(j + 1 < M)` `                ``sum += mat[i, j + 1];`   `            ``// If the sum is a prime number` `            ``if` `(prime[sum])` `                ``count++;` `        ``}` `    ``}`   `    ``// Return the count` `    ``return` `count;` `}`   `// Driver Code` `public` `static` `void` `Main(String[] args)` `{` `    ``SieveOfEratosthenes();`   `    ``int` `[,]mat = { { 1, 2, 3, 4 },` `                   ``{ 5, 6, 7, 8 },` `                   ``{ 9, 10, 11, 12 } };`   `    ``// Function call` `    ``Console.Write(PrimeSumCells(mat) + ``"\n"``);` `}` `}`   `// This code is contributed by sapnasingh4991`

Output:
```6

```

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Improved By : sapnasingh4991, chitranayal

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