<< problem 31 - Coin sums | Digit cancelling fractions - problem 33 >> |

# Problem 32: Pandigital products

(see projecteuler.net/problem=32)

We shall say that an n-digit number is pandigital if it makes use of all the digits 1 to n exactly once;

for example, the 5-digit number, 15234, is 1 through 5 pandigital.

The product 7254 is unusual, as the identity, 39 * 186 = 7254, containing multiplicand, multiplier, and product is 1 through 9 pandigital.

Find the sum of all products whose multiplicand/multiplier/product identity can be written as a 1 through 9 pandigital.

*HINT:* Some products can be obtained in more than one way so be sure to only include it once in your sum.

# My Algorithm

We have to solve an equation a * b = c.

There are 9!=362880 permutations of `{ 1,2,3,4,5,6,7,8,9 }`

and for each of those permutation I try all possible combinations of a, b and c.

All digits (0..9) are stored in a plain `std::vector`

and the STL's function `std::next_permutation`

generates all permutations.

Then two nested loops split the sequence into three parts (a has length `lenA`

, b has length `lenB`

and c has length `lenC`

).

If a * b = c then my program stores the product c in `std::set`

named `valid`

. Duplicates are automatically avoided by the `std::set`

.

Finally all elements of that `std::set`

are added.

## Modifications by HackerRank

The digits are 1 to n instead of 1 to 9.

## Note

There are many opportunities for optimizations but the code is already extremely fast.

# Interactive test

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

This is equivalent to`echo 5 | ./32`

Output:

*Note:* the original problem's input `9`

__cannot__ be entered

because just copying results is a soft skill reserved for idiots.

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

# My code

… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too.

#include <iostream>
#include <vector>
#include <set>
#include <algorithm>
int main()
{
// read highest digit
unsigned int maxDigit;
std::cin >> maxDigit;
// all digits from 1..9
std::vector<unsigned int> digits = { 1,2,3,4,5,6,7,8,9 };
// remove higher numbers so there is only 1..n left
digits.resize(maxDigit);
// all pandigital products
std::set<unsigned int> valid;
// create all permutations
do
{
// let's say a * b = c
// each variable contains at least one digit
// the sum of their digits is limited by n (which should be 9)
// try all combinations of lengths with the current permutation of digits
for (unsigned int lenA = 1; lenA < maxDigit; lenA++)
for (unsigned int lenB = 1; lenB < maxDigit - lenA; lenB++)
{
unsigned int lenC = maxDigit - lenA - lenB;
// a*b=c => c>=a && c>=b => c has at least as many digits as a or b
if (lenC < lenA || lenC < lenB)
break;
// pos contains the currently used position in "digits"
unsigned int pos = 0;
// build "a" out of the first digits
unsigned int a = 0;
for (unsigned int i = 1; i <= lenA; i++)
{
a *= 10;
a += digits[pos++];
}
// next digits represent "b"
unsigned int b = 0;
for (unsigned int i = 1; i <= lenB; i++)
{
b *= 10;
b += digits[pos++];
}
// and the same for "c"
unsigned int c = 0;
for (unsigned int i = 1; i <= lenC; i++)
{
c *= 10;
c += digits[pos++];
}
// is a*b = c ?
if (a*b == c)
valid.insert(c);
}
} while (std::next_permutation(digits.begin(), digits.end()));
// find sum
unsigned int sum = 0;
for (auto x : valid)
sum += x;
std::cout << sum << std::endl;
return 0;
}

This solution contains 12 empty lines, 16 comments and 4 preprocessor commands.

# Benchmark

The correct solution to the original Project Euler problem was found in 0.05 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

April 6, 2017 added comments

# Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler032

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 rarely an option.

# Links

projecteuler.net/thread=32 - **the** best forum on the subject (*note:* you have to submit the correct solution first)

Code in various languages:

C#: www.mathblog.dk/project-euler-32-pandigital-products/ (written by Kristian Edlund)

Haskell: github.com/nayuki/Project-Euler-solutions/blob/master/haskell/p032.hs (written by Nayuki)

Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p032.java (written by Nayuki)

Mathematica: github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p032.mathematica (written by Nayuki)

C: github.com/eagletmt/project-euler-c/blob/master/30-39/problem32.c (written by eagletmt)

Go: github.com/frrad/project-euler/blob/master/golang/Problem032.go (written by Frederick Robinson)

Javascript: github.com/dsernst/ProjectEuler/blob/master/32 Pandigital products.js (written by David Ernst)

Scala: github.com/samskivert/euler-scala/blob/master/Euler032.scala (written by Michael Bayne)

Those links are just an unordered selection of source code I found with a semi-automatic search script on Google/Bing/GitHub/whatever.

You will probably stumble upon better solutions when searching on your own. Maybe not all linked resources produce the correct result.

# Heatmap

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

green | solutions solve the original Project Euler problem and have a perfect score of 100% at Hackerrank, too | |

yellow | solutions score less than 100% at Hackerrank (but still solve the original problem easily) | |

gray | problems are already solved but I haven't published my solution yet | |

blue | solutions are relevant for Project Euler only: there wasn't a Hackerrank version of it (at the time I solved it) or it differed too much | |

orange | problems are solved but exceed the time limit of one minute or the memory limit of 256 MByte | |

red | problems are not solved yet but I wrote a simulation to approximate the result or verified at least the given example - usually I sketched a few ideas, too |

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I scored 13,386 points (out of 15600 possible points, top rank was 17 out of ≈60000 in August 2017) at Hackerrank's Project Euler+.

My username at Project Euler is

**stephanbrumme**while it's stbrumme at Hackerrank.

Look at my progress and performance pages to get more details.

# Copyright

I hope you enjoy my code and learn something - or give me feedback how I can improve my solutions.

All of my solutions can be used for any purpose and I am in no way liable for any damages caused.

You can even remove my name and claim it's yours. But then you shall burn in hell.

The problems and most of the problems' images were created by Project Euler.

Thanks for all their endless effort !!!

<< problem 31 - Coin sums | Digit cancelling fractions - problem 33 >> |