<< problem 42 - Coded triangle numbers | Pentagon numbers - problem 44 >> |

# 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.

# My 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:

This is equivalent to`echo 8 | ./43`

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)*

# Benchmark

The correct solution to the original Project Euler problem was found in 0.4 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 solves **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 rarely 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 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.

red problems are solved but exceed the time limit of one minute or the memory limit of 256 MByte.

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

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I scored 13,183 points (out of 15300 possible points, top rank was 17 out of ≈60000 in August 2017) at Hackerrank's Project Euler+.

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

My username at Project Euler is

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

# 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 42 - Coded triangle numbers | Pentagon numbers - problem 44 >> |