<< 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.

# 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 **0.42** 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.

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

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 |

26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 |

51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 | 61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 | 70 | 71 | 72 | 73 | 74 | 75 |

76 | 77 | 78 | 79 | 80 | 81 | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 | 90 | 91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | 100 |

101 | 102 | 103 | 104 | 105 | 106 | 107 | 108 | 109 | 110 | 111 | 112 | 113 | 114 | 115 | 116 | 117 | 118 | 119 | 120 | 121 | 122 | 123 | 124 | 125 |

126 | 127 | 128 | 129 | 130 | 131 | 132 | 133 | 134 | 135 | 136 | 137 | 138 | 139 | 140 | 141 | 142 | 143 | 144 | 145 | 146 | 147 | 148 | 149 | 150 |

151 | 152 | 153 | 154 | 155 | 156 | 157 | 158 | 159 | 160 | 161 | 162 | 163 | 164 | 165 | 166 | 167 | 168 | 169 | 170 | 171 | 172 | 173 | 174 | 175 |

176 | 177 | 178 | 179 | 180 | 181 | 182 | 183 | 184 | 185 | 186 | 187 | 188 | 189 | 190 | 191 | 192 | 193 | 194 | 195 | 196 | 197 | 198 | 199 | 200 |

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