<< problem 10 - Summation of primes Highly divisible triangular number - problem 12 >>

# Problem 11: Largest product in a grid

In the 20x20 grid below, four numbers along a diagonal line have been marked in red.

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 product of these numbers is 26 x 63 x 78 x 14 = 1788696.

What is the greatest product of four adjacent numbers in the same direction
(up, down, left, right, or diagonally) in the 20x20 grid?

# Algorithm

For each position of the grid I find the product of 4 connected cells:
1. current cell and its three neighbors when going to the right side
2. current cell and its three neighbors below it
3. current cell and its three neighbors going right and down
4. current cell and its three neighbors going left and down

For each of these steps I have to check whether enough neighbor exist.
And finally the greatest product is printed.

## Note

Be careful when reading a 2D matrix from console:
the outer loop must belong to the y-axis, the inner to x.
A common mistake of mine is to swap those two.

# My code

… was written in C++ and can be compiled with G++, Clang++, Visual C++. You can download it, as well as the input data, too.

       #include <iostream>

int main()
{
// always a 20x20 matrix
const unsigned int Size = 20;
unsigned int matrix[Size][Size];

for (unsigned int y = 0; y < Size; y++)
for (unsigned int x = 0; x < Size; x++)
std::cin >> matrix[x][y];

unsigned int best = 0;
// walk through all cells of the matrix
for (unsigned int y = 0; y < Size; y++)
for (unsigned int x = 0; x < Size; x++)
{
// three more horizontal cells (right)
if (x + 3 < Size)
{
unsigned int current = matrix[x][y] * matrix[x+1][y] * matrix[x+2][y] * matrix[x+3][y];
if (best < current)
best = current;
}
// three more vertical cells available (down)
if (y + 3 < Size)
{
unsigned int current = matrix[x][y] * matrix[x][y+1] * matrix[x][y+2] * matrix[x][y+3];
if (best < current)
best = current;
}
// three more diagonal cells (right-down)
if (x + 3 < Size && y + 3 < Size)
{
unsigned int current = matrix[x][y] * matrix[x+1][y+1] * matrix[x+2][y+2] * matrix[x+3][y+3];
if (best < current)
best = current;
}
// three more diagonal cells (left-down)
if (x + 3 < Size && y >= 3)
{
unsigned int current = matrix[x][y] * matrix[x+1][y-1] * matrix[x+2][y-2] * matrix[x+3][y-3];
if (best < current)
best = current;
}
}

std::cout << best << std::endl;
return 0;
}


This solution contains 4 empty lines, 7 comments and 1 preprocessor command.

# Interactive test

You can submit your own input to my program and it will be instantly processed at my server:

Input data (separated by spaces or newlines):

This is equivalent to
echo "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" | ./11

Output:

(this interactive test is still under development, computations will be aborted after one second)

# Benchmark

The correct solution to the original Project Euler problem was found in less than 0.01 seconds on a Intel® Core™ i7-2600K CPU @ 3.40GHz.
(compiled for x86_64 / Linux, GCC flags: -O3 -march=native -fno-exceptions -fno-rtti -std=c++11 -DORIGINAL)

See here for a comparison of all solutions.

Note: interactive tests run on a weaker (=slower) computer. Some interactive tests are compiled without -DORIGINAL.

# Changelog

February 23, 2017 submitted solution

# Hackerrank

My code solves 6 out of 6 test cases (score: 100%)

# Difficulty

Project Euler ranks this problem at 5% (out of 100%).

Hackerrank describes this problem as easy.

Note:
Hackerrank has strict execution time limits (typically 2 seconds for C++ code) and often a much wider input range than the original problem.
In my opinion, Hackerrank's modified problems are usually a lot harder to solve. As a rule thumb: brute-force is never an option.

projecteuler.net/thread=11 - the best forum on the subject (note: you have to submit the correct solution first)

Code in various languages:

Python: www.mathblog.dk/greatest-product-in-20x20-grid/ (written by Kristian Edlund)
Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p011.java (written by Nayuki)
Mathematica: github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p011.mathematica (written by Nayuki)
C: github.com/eagletmt/project-euler-c/blob/master/10-19/problem11.c (written by eagletmt)
Go: github.com/frrad/project-euler/blob/master/golang/Problem011.go (written by Frederick Robinson)
Javascript: github.com/dsernst/ProjectEuler/blob/master/11 Largest product in a grid.js (written by David Ernst)
Scala: github.com/samskivert/euler-scala/blob/master/Euler011.scala (written by Michael Bayne)

# Heatmap

green problems solve the original Project Euler problem and have a perfect score of 100% at Hackerrank, too.
yellow problems score less than 100% at Hackerrank (but still solve the original problem).
gray problems are already solved but I haven't published my solution yet.
blue problems are solved and there wasn't a Hackerrank version of it at the time I solved it or I didn't care about it because it differed too much.

Please click on a problem's number to open my solution to that problem:

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The 163 solved problems had an average difficulty of 22.2% at Project Euler and I scored 11,907 points (out of 13200) at Hackerrank's Project Euler+.
My username at Project Euler is stephanbrumme while it's stbrumme at Hackerrank.
 << problem 10 - Summation of primes Highly divisible triangular number - problem 12 >>
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