# Project Euler

### A Taste of Number Theory

#### Problem 11: Largest Product in a Grid

In the 20 x 20 grid below, four numbers along a diagonal line have been marked in red. The product of these numbers is 26 * 63 * 78 * 14 = 1788696. What is the greatest product of four adjacent numbers in the same direction (up, down, left, right, or diagonally) in the 20 x 20 grid?

08 02 22 97 38 15 00 40 00 75 04 05 07 78 52 12 50 77 91 08
49 49 99 40 17 81 18 57 60 87 17 40 98 43 69 48 04 56 62 00
81 49 31 73 55 79 14 29 93 71 40 67 53 88 30 03 49 13 36 65
52 70 95 23 04 60 11 42 69 24 68 56 01 32 56 71 37 02 36 91
22 31 16 71 51 67 63 89 41 92 36 54 22 40 40 28 66 33 13 80
24 47 32 60 99 03 45 02 44 75 33 53 78 36 84 20 35 17 12 50
32 98 81 28 64 23 67 10 26 38 40 67 59 54 70 66 18 38 64 70
67 26 20 68 02 62 12 20 95 63 94 39 63 08 40 91 66 49 94 21
24 55 58 05 66 73 99 26 97 17 78 78 96 83 14 88 34 89 63 72
21 36 23 09 75 00 76 44 20 45 35 14 00 61 33 97 34 31 33 95
78 17 53 28 22 75 31 67 15 94 03 80 04 62 16 14 09 53 56 92
16 39 05 42 96 35 31 47 55 58 88 24 00 17 54 24 36 29 85 57
86 56 00 48 35 71 89 07 05 44 44 37 44 60 21 58 51 54 17 58
19 80 81 68 05 94 47 69 28 73 92 13 86 52 17 77 04 89 55 40
04 52 08 83 97 35 99 16 07 97 57 32 16 26 26 79 33 27 98 66
88 36 68 87 57 62 20 72 03 46 33 67 46 55 12 32 63 93 53 69
04 42 16 73 38 25 39 11 24 94 72 18 08 46 29 32 40 62 76 36
20 69 36 41 72 30 23 88 34 62 99 69 82 67 59 85 74 04 36 16
20 73 35 29 78 31 90 01 74 31 49 71 48 86 81 16 23 57 05 54
01 70 54 71 83 51 54 69 16 92 33 48 61 43 52 01 89 19 67 48

#### The Catch

This problem is fairly straightforward; there is no other way but to check all routes of the 20 x 20 grid.

#### The Light

Checking columns and rows are simple: just loop through each row and column and find the product of every 4 entries. Checking 2 left and right diagonals require more check points to prevent out of bound exception.

#### The Code

```public class Problem11
{
public static void main( String[] args )
{
int[][] a = new int[20][20];
int[] a1d = { 8, 02, 22, 97, 38, 15, 00, 40, 00, 75, 04, 05, 07, 78, 52, 12, 50, 77, 91,  8,
49, 49, 99, 40, 17, 81, 18, 57, 60, 87, 17, 40, 98, 43, 69, 48, 04, 56, 62, 00,
81, 49, 31, 73, 55, 79, 14, 29, 93, 71, 40, 67, 53, 88, 30, 03, 49, 13, 36, 65,
52, 70, 95, 23, 04, 60, 11, 42, 69, 24, 68, 56, 01, 32, 56, 71, 37, 02, 36, 91,
22, 31, 16, 71, 51, 67, 63, 89, 41, 92, 36, 54, 22, 40, 40, 28, 66, 33, 13, 80,
24, 47, 32, 60, 99, 03, 45, 02, 44, 75, 33, 53, 78, 36, 84, 20, 35, 17, 12, 50,
32, 98, 81, 28, 64, 23, 67, 10, 26, 38, 40, 67, 59, 54, 70, 66, 18, 38, 64, 70,
67, 26, 20, 68, 02, 62, 12, 20, 95, 63, 94, 39, 63,  8, 40, 91, 66, 49, 94, 21,
24, 55, 58, 05, 66, 73, 99, 26, 97, 17, 78, 78, 96, 83, 14, 88, 34, 89, 63, 72,
21, 36, 23,  9, 75, 00, 76, 44, 20, 45, 35, 14, 00, 61, 33, 97, 34, 31, 33, 95,
78, 17, 53, 28, 22, 75, 31, 67, 15, 94, 03, 80, 04, 62, 16, 14,  9, 53, 56, 92,
16, 39, 05, 42, 96, 35, 31, 47, 55, 58, 88, 24, 00, 17, 54, 24, 36, 29, 85, 57,
86, 56, 00, 48, 35, 71, 89, 07, 05, 44, 44, 37, 44, 60, 21, 58, 51, 54, 17, 58,
19, 80, 81, 68, 05, 94, 47, 69, 28, 73, 92, 13, 86, 52, 17, 77, 04, 89, 55, 40,
04, 52,  8, 83, 97, 35, 99, 16, 07, 97, 57, 32, 16, 26, 26, 79, 33, 27, 98, 66,
88, 36, 68, 87, 57, 62, 20, 72, 03, 46, 33, 67, 46, 55, 12, 32, 63, 93, 53, 69,
04, 42, 16, 73, 38, 25, 39, 11, 24, 94, 72, 18,  8, 46, 29, 32, 40, 62, 76, 36,
20, 69, 36, 41, 72, 30, 23, 88, 34, 62, 99, 69, 82, 67, 59, 85, 74, 04, 36, 16,
20, 73, 35, 29, 78, 31, 90, 01, 74, 31, 49, 71, 48, 86, 81, 16, 23, 57, 05, 54,
01, 70, 54, 71, 83, 51, 54, 69, 16, 92, 33, 48, 61, 43, 52, 01, 89, 19, 67, 48,};
int index = 0;
int max = 0;
int product = 0;

for(int x = 0; x < a.length; x++)
{
for(int y = 0; y < a.length; y++)
{
a[x][y] = a1d[index++];
}
}

//Check columns
int x3 = 0;
for(int y = 0; y < a.length; y++)
{
while(x3 < a.length - 3)
{
product = a[x3][y] *
a[x3 + 1][y] *
a[x3 + 2][y] *
a[x3 + 3][y];
if(product > max)
max = product;

++x3;
}
x3 = 0;
}

//Check rows
int y4 = 0;
for( int x = 0; x < a.length; x++)
{
while(y4 < a.length - 3)
{
product = a[x][y4] *
a[x][y4 + 1] *
a[x][y4 + 2] *
a[x][y4 + 3];

if(product > max)
max = product;

++y4;
}
y4 = 0;
}

//Check left diagonal below
int x = 1;
int tmp = 1;
while(x < a.length - 3)
{
for(int y = 0; y < a.length - 3 - tmp; y++)
{
product = a[x][y] *
a[x + 1][y + 1] *
a[x + 1][y + 2] *
a[x + 1][y + 3];

if(product > max)
max = product;

++x;
}
x = ++tmp;
}

//Chek left diagonal above
int x2 = 1;
int tmp2 = 1;
while(x2 < a.length - 3)
{
for(int y = 0; y < a.length - 3 - tmp2; y++)
{
product = a[y][x2] *
a[y + 1][x2 + 1] *
a[y + 2][x2 + 2] *
a[y + 3][x2 + 3];

if(product > max)
max = product;

++x2;
}
x2 = ++tmp2;
}

//Check left diagonal middle line
for(int i = 0; i < a.length - 3; i++)
{
product = a[i][i] *
a[i + 1][i + 1] *
a[i + 2][i + 2] *
a[i + 3][i + 3];

if(product > max)
max = product;
}

//Check right diagonal above and diagonal middle line
int x5 = 3;
int tmp5 = 3;
int count = 16;
while(x5 < a.length)
{
for(int y = 0; y < a.length - 3 - count; y++)
{
product = a[x5][y] *
a[x5 - 1][y + 1] *
a[x5 - 2][y + 2] *
a[x5 - 3][y + 3];

if(product > max)
max = product;

x5--;
}
count--;
x5 = ++tmp5;
}

//Check right diagonal below
int y6 = 1;
int tmp6 = 1;
while(y6 < a.length - 3)
{
for(x = a.length - 1; x > 3 && y6 < a.length - 3; x--)
{
product = a[x][y6] *
a[x - 1][y6 + 1] *
a[x - 2][y6 + 2] *
a[x - 3][y6 + 3];

if(product > max)
max = product;

y6++;
}
y6 = ++tmp6;
}
System.out.println(max);
}
}
```