Largest product in a grid

Project Euler problem 11 says:


In the 20×20 grid below, four numbers along a diagonal line have been marked in bold. [red in the original]


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 102638 40 67 59 54 70 66 18 38 64 70

67 26 20 68 02 62 12 20 956394 39 63 08 40 91 66 49 94 21

24 55 58 05 66 73 99 26 97 177878 96 83 14 88 34 89 63 72

21 36 23 09 75 00 76 44 20 45 351400 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 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 20×20 grid?

So, I coded this:

```
grid=[8,2,22,97,38,15,0,40,0,75,4,5,7,78,52,12,50,77,91,8,
49,49,99,40,17,81,18,57,60,87,17,40,98,43,69,48,4,56,62,0,
81,49,31,73,55,79,14,29,93,71,40,67,53,88,30,3,49,13,36,65,
52,70,95,23,4,60,11,42,69,24,68,56,1,32,56,71,37,2,36,91,
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,

Your code would be much simpler if you'd use a list of lists for the matrix:

grid = [
  [...],
  [...],
  ...
]

each row is a list itself, so to get the element at (x, y) we can now say grid[y][x]. This is massively more readable than your previous grid[y*20+x].

Your code now looks like:

#horizontal and vertical
for h in xrange(0,20):
    for hsub in xrange(0,17):
        if grid[h][hsub]*grid[h][hsub+1]*grid[h][hsub+2]*grid[h][hsub+3] > largest[0]:
            largest[0] = grid[h][hsub]*grid[h][hsub+1]*grid[h][hsub+2]*grid[h][hsub+3]
        if grid[hsub][h]* grid[hsub+1][h]* grid[hsub+2][h]* grid[hsub+3][h] > largest[1]:
            largest[1] = grid[hsub][h]* grid[hsub+1][h]* grid[hsub+2][h]* grid[hsub+3][h]

#diagonal right and left
for r in xrange(0,17):
    for rsub in xrange (0,17):
        if grid[rsub][r]* grid[rsub+1][1+r]* grid[rsub+2][2+r]* grid[rsub+3][3+r] > largest[2]:
            largest[2] = grid[rsub][r]* grid[rsub+1][1+r]* grid[rsub+2][2+r]* grid[rsub+3][3+r] 
        if grid[rsub][3+r]* grid[rsub+1][2+r]* grid[rsub+2][1+r]* grid[rsub+3][r] > largest[3]:
            largest[3] =  grid[rsub][3+r]* grid[rsub+1][2+r]* grid[rsub+2][1+r]* grid[rsub+3][r]

At this step I also removed uneccessary empty lines from your code.

We see that each expression for summing adjacent cells in a certain direction is repeated two times. We can avoid this by using a variable each time:

#horizontal and vertical
for h in xrange(0,20):
    for hsub in xrange(0,17):
        horizontal = grid[h][hsub] * grid[h][hsub+1] * grid[h][hsub+2] * grid[h][hsub+3]
        vertical   = grid[hsub][h] * grid[hsub+1][h] * grid[hsub+2][h] * grid[hsub+3][h]
        if horizontal > largest[0]:
            largest[0] = horizontal
        if vertical > largest[1]:
            largest[1] = vertical

#diagonal right and left
for r in xrange(0,17):
    for rsub in xrange (0,17):
        right_diagonal = grid[rsub][0+r] * grid[rsub+1][1+r] * grid[rsub+2][2+r] * grid[rsub+3][3+r]
        left_diagonal  = grid[rsub][3+r] * grid[rsub+1][2+r] * grid[rsub+2][1+r] * grid[rsub+3][r]
        if right_diagonal > largest[2]:
            largest[2] = right_diagonal
        if left_diagonal > largest[3]:
            largest[3] = left_diagonal

This format makes it much easier to spot errors – just as I'm writing this, I see that we forgot a +1 in the left diagonal (corrected in the previous code as well).

All of these products look similar: We have some starting point and a direction. We then take three steps in that direction, and multiply all numbers. We could abstract this:

def product_in_direction(grid, start, direction, steps):
    x0, y0 = start
    dx, dy = direction
    product = 1
    for n in range(steps):
        product *= grid[y0 + n*dy][x0 + n*dx]
    return product

Now the rest of the code simplifies to:

#horizontal and vertical
for h in xrange(0,20):
    for hsub in xrange(0,17):
        horizontal = product_in_direction(grid, (hsub, h), (1, 0), 4)
        vertical   = product_in_direction(grid, (h, hsub), (0, 1), 4)
        if horizontal > largest[0]:
            largest[0] = horizontal
        if vertical > largest[1]:
            largest[1] = vertical

#diagonal right and left
for r in xrange(0,17):
    for rsub in xrange (0,17):
        right_diagonal = product_in_direction(grid, (rsub, r  ), (1,  1), 4)
        left_diagonal  = product_in_direction(grid, (rsub, r+3), (1, -1), 4)
        if right_diagonal > largest[2]:
            largest[2] = right_diagonal
        if left_diagonal > largest[3]:
            largest[3] = left_diagonal

As a next step, we will remove the unnecesary largest array: Why do we need to remember four different values when we only want the largest?

largest = 0

#horizontal and vertical
for h in xrange(0,20):
    for hsub in xrange(0,17):
        largest = max(
            product_in_direction(grid, (hsub, h), (1, 0), 4), # horizontal
            product_in_direction(grid, (h, hsub), (0, 1), 4), # vertical
            largest,
        )

#diagonal right and left
for r in xrange(0,17):
    for rsub in xrange (0,17):
        largest = max(
            product_in_direction(grid, (rsub, r  ), (1,  1), 4), # right diagonal
            product_in_direction(grid, (rsub, r+3), (1, -1), 4), # left diagonal
            largest,
         )

It is annoying that we iterate over two different ranges. If we update our product_in_direction function to employ a range check, then we can avoid the need for that:

``
def product_in_direction(grid, start, direction, steps):
x0, y0 = start
dx, dy = direction

if not(0 <= y0 < len(grid) and
0 <= y0 + (steps - 1)*dy < len(grid) and
0 <= x0 < len(grid[y0]) and
0 <= x0 + (steps - 1)*dx < len(grid[y0])):
return 0

product = 1
for n in range(steps):
product = grid[y0 + ndy][x0 + n*dx]
return product


StackExchange Code Review Q#37767, answer score: 13