<< 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.
Interactive test
You can submit your own input to my program and it will be instantly processed at my server:
This is equivalent toecho 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)
My code
… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too. Or just jump to my GitHub repository.
#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.
Benchmark
The correct solution to the original Project Euler problem was found in 0.4 seconds on an Intel® Core™ i7-2600K CPU @ 3.40GHz.
(compiled for x86_64 / Linux, GCC flags: -O3 -march=native -fno-exceptions -fno-rtti -std=gnu++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:
C# www.mathblog.dk/project-euler-43-pandigital-numbers-sub-string-divisibility/ (written by Kristian Edlund)
C github.com/eagletmt/project-euler-c/blob/master/40-49/problem43.cc (written by eagletmt)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p043.java (written by Nayuki)
Javascript github.com/dsernst/ProjectEuler/blob/master/43 Sub-string divisibility.js (written by David Ernst)
Go github.com/frrad/project-euler/blob/master/golang/Problem043.go (written by Frederick Robinson)
Mathematica github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p043.mathematica (written by Nayuki)
Scala github.com/samskivert/euler-scala/blob/master/Euler043.scala (written by Michael Bayne)
Perl github.com/gustafe/projecteuler/blob/master/043-Substring-divisibility.pl (written by Gustaf Erikson)
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 and/or exceed time/memory limits.
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 | |
black | problems are solved but access to the solution is blocked for a few days until the next problem is published | |
[new] | the flashing problem is the one I solved most recently |
I stopped working on Project Euler problems around the time they released 617.
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I scored 13526 points (out of 15700 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 42 - Coded triangle numbers | Pentagon numbers - problem 44 >> |