Problem 43: Sub-string divisibility

(see projecteuler.net/problem=43)

The number, 1406357289, is a 0 to 9 pandigital number because it is made up of each of the digits 0 to 9 in some order,
but it also has a rather interesting sub-string divisibility property.

Let d_1 be the 1st digit, d_2 be the 2nd digit, and so on. In this way, we note the following:

d_2 d_3 d_4 = 406 is divisible by 2
d_3 d_4 d_5 = 063 is divisible by 3
d_4 d_5 d_6 = 635 is divisible by 5
d_5 d_6 d_7 = 357 is divisible by 7
d_6 d_7 d_8 = 572 is divisible by 11
d_7 d_8 d_9 = 728 is divisible by 13
d_8 d_9 d_10 = 289 is divisible by 17

Find the sum of all 0 to 9 pandigital numbers with this property.

Algorithm

Once more, std::next_permutation turns out to be a rather handy feature:
I generate all permutations of pan = "0123456789" and check all its substrings with length 3 for divisibility with the first prime numbers.

str2num converts an ASCII string to a number, ignoring leading zeros: str2num("012") = 12

We need only the first 7 prime numbers - that's why I opted against a full prime sieve and just declared a precomputed array primes.

Modifications by HackerRank

The number of digits can vary. All I do is adjusting pan: anything else remains the same.
Hackerrank accepts pandigital numbers that start with a zero.

Note

Ten digits can exceed 32 bits, therefore you'll find unsigned long long instead of unsigned int in a few places.

There are many optimization possible: when applying the rules of divisibility by 3, 5, ... then
we could easily exclude the majority of permutations. Nevertheless, the non-optimized code runs fast enough.

My code

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

#include <string>
#include <iostream>
#include <algorithm>
 
// convert a string to a number
unsigned long long str2num(const std::string& x)
{
// process string from left to right
unsigned long long result = 0;
for (auto c : x)
{
// shift digits
result *= 10;
// add new digit on the right-hand side
result += c - '0'; // was ASCII
}
return result;
}
 
int main()
{
// available digits
std::string pan = "0123456789"; // unlike other problems, zero is allowed this time
 
// remove a few digits if test case requires this
unsigned int maxDigit;
std::cin >> maxDigit;
pan.erase(maxDigit + 1);
 
// all divisors
const unsigned int primes[] = { 2,3,5,7,11,13,17 };
 
// result
unsigned long long sum = 0;
 
// look at all permutations
do
{
// let's assume it's a great number ;-)
bool ok = true;
 
// check each 3-digit substring for divisibility
for (unsigned int i = 0; i + 2 < maxDigit; i++)
{
// check pan[1..3] against primes[0],
// check pan[2..4] against primes[1],
// check pan[3..5] against primes[2] ...
std::string check = pan.substr(i + 1, 3);
if (str2num(check) % primes[i] != 0)
{
// nope ...
ok = false;
break;
}
}
 
// passed all tests, it's great indeed !
if (ok)
sum += str2num(pan);
} while (std::next_permutation(pan.begin(), pan.end()));
 
std::cout << sum << std::endl;
return 0;
}

This solution contains 9 empty lines, 16 comments and 3 preprocessor commands.

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 8 | ./43

Output:

(please click 'Go !')

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)

Benchmark

The correct solution to the original Project Euler problem was found in 0.42 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 19, 2017 added comments

Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler043

My code solved 7 out of 7 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.

Links

projecteuler.net/thread=43 - the best forum on the subject (note: you have to submit the correct solution first)

Code in various languages:

Python: www.mathblog.dk/project-euler-43-pandigital-numbers-sub-string-divisibility/ (written by Kristian Edlund)
Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p043.java (written by Nayuki)
Mathematica: github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p043.mathematica (written by Nayuki)
C: github.com/eagletmt/project-euler-c/blob/master/40-49/problem43.cc (written by eagletmt)
Go: github.com/frrad/project-euler/blob/master/golang/Problem043.go (written by Frederick Robinson)
Javascript: github.com/dsernst/ProjectEuler/blob/master/43 Sub-string divisibility.js (written by David Ernst)
Scala: github.com/samskivert/euler-scala/blob/master/Euler043.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 already 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.

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The 126 solved problems had an average difficulty of 16.0% at Project Euler and I scored 11,074 points (out of 12500) at Hackerrank's Project Euler+.
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