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Count Magic squares in a grid in C++
We are given with a matrix of numbers. The goal is to find the number of Magic squares present inside the given matrix.
A Magic Square, if taken as a matrix, is a 3X3 matrix which contains the elements ranging from 1 to 9 just like a grid in Sudoku. Properties are −
- All numbers occur exactly once.
- Sum of all 9 cells in the matrix is 45.
- Sum of each row of 3 is 15.
- Sum of each column of 3 is 15.
- Sum of diagonals of 3 is 5.
- To get such sums, 5 is always in the middle of both diagonals.
Input
int arr[][]= { { 1,2,3,0 }, { 4,5,6,1 }, { 7,8,9,0 } };Output
Magic Squares present: 0
Explanation − Making a matrix form to understand −
| 1 | 2 | 3 | 0 |
| 4 | 5 | 6 | 1 |
| 7 | 8 | 9 | 0 |
All elements are between 1 to 9 and are unique. But,
1+2+3=6 != 4+5+6=15 != 7+8+9=23
Also diagonals sum is not equal to 15.
Input
arr[][]= { { 1,2,3,0 }, { 4,5,6,1 }, { 7,8,9,0 } };Output
Magic Squares present : 1
Explanation − Making a matrix form to understand −
| 1 | 6 | 8 | 4 |
| 8 | 1 | 6 | 0 |
| 3 | 5 | 7 | 1 |
| 4 | 9 | 2 | 0 |
All numbers are unique and in the range 1 to 9.
Rows sum, 8+1+6=3+5+7=4+9+2=15
Columns sum, 8+3+4=1+5+9=6+7+2=15
Diagonals sum, 8+5+2=4+5+6=15
Also adding 1 to 9 we get 45, and 5 is middle of both diagonals.
Approach used in the below program is as follows
Integer array Grid[][] stores the numbers, row and col stores its dimensions.
Function magicSquare(int a, int b…..int i) takes all 9 elements as input and checks if it makes the magic square. Returns 1 if it is magic square, returns 1 otherwise.
Here we will take array arr[9] to store all parameters and check if they are unique by checking that they occupy unique cells in it. If a cell has count<1 means no. is not in range of 1 to 9, or if count>1 then no. is not unique. set flag=0
If flag is 1 then we will check for row, column, diagonal sums as 15. If true then it is a magic square.Return 1 else return 0.
Function countSquares (int G[3][4],int R, int C) takes a grid, its rows and columns as input and counts the no. of magic squares present in it.
Count is used to store count of the number of such squares.
Start traversing from first element till row-2, col-2 ( 3X3 matrix )
If the middle of diagonal G[i+1][j+1] is not 5 then magic square is not possible, skip the current iteration.
Else check for all 9 elements by passing it to magicSquare( int a to i )
All 9 elements including G[i][j] are G[i+1][j], G[i+2][j], G[i+1][j+1],G[i][j+1],G[i][j+2],G[i+2][j+2],G[i+2][j+1], G[i+1][j+2]
If it returns 1 increase count.
In the end return count as desired result.
Example
#include <bits/stdc++.h>
using namespace std;
// to check is subgrid is Magic Square
int magicSquare(int a, int b, int c, int d, int e,
int f, int g, int h, int i){
int flag=1; // to mark all numbers are unique and between 1 to 9
int arr[9]={0};
arr[a-1]++;
arr[b-1]++;
arr[c-1]++;
arr[d-1]++;
arr[e-1]++;
arr[f-1]++;
arr[g-1]++;
arr[h-1]++;
arr[i-1]++;
for(int i=0;i<9;i++)
if(arr[i]>1 || arr[i]<1) //every number occurs exactly once{
flag=0;
break;
}
// checking all sums as 15
if (flag==1 && (a + b + c) == 15 && (d + e + f) == 15 && (g + h + i) == 15 && (a + d + g) == 15 &&(b + e + h) == 15 && (c + f + i) == 15 &&(a + e + i) == 15 && (c + e + g) == 15)
return 1;
return 0;
}
int countSquares(int G[3][4],int R, int C){
int count = 0;
for (int i = 0; i < R - 2; i++)
for (int j = 0; j < C - 2; j++) {
if (G[i + 1][j + 1] != 5)
continue;
int ismagic=magicSquare(G[i][j], G[i][j + 1], G[i][j + 2], G[i + 1][j], G[i + 1][j + 1],
G[i + 1][j + 2], G[i + 2][j], G[i + 2][j + 1], G[i + 2][j + 2]);
// check for magic square subgrid
if (ismagic==1)
count++;
}
return count;
}
int main(){
int Grid[3][4] = { { 4, 3, 8, 4 },{ 9, 5, 1, 9 },{ 2, 7, 6, 2 } };
int row=3;
int col=4;
cout <<"Count of Magic Squares in Grid: "<<countSquares(Grid,row,col);
return 0;
}Output
Count of Magic Squares in Grid: 1
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