<< problem 169 - Exploring the number of different ways a number ... | Finding numbers for which the sum of the squares ... - problem 171 >> |
Problem 170: Find the largest 0 to 9 pandigital that can be formed by concatenating products
(see projecteuler.net/problem=170)
Take the number 6 and multiply it by each of 1273 and 9854:
6 * 1273 = 7638
6 * 9854 = 59124
By concatenating these products we get the 1 to 9 pandigital 763859124. We will call 763859124 the "concatenated product of 6 and (1273,9854)". Notice too, that the concatenation of the input numbers, 612739854, is also 1 to 9 pandigital.
The same can be done for 0 to 9 pandigital numbers.
What is the largest 0 to 9 pandigital 10-digit concatenated product of an integer with two or more other integers, such that the concatenation of the input numbers is also a 0 to 9 pandigital 10-digit number?
My Algorithm
I analyze all pandigital numbers, beginning with the largest (9876543210
) and slowly decrement towards 1023456789
.
The easiest way is to store the pandigital number in a std::string
and call std::prev_permutation
repeatedly (see my variable current
).
A bold assumption is that the pandigital number only needs to be split into two parts left and right such that
left = factor * one and right = factor * two. If concat(factor, one, two) is 10-pandigital then we found the result.
Neither left nor right can begin with a zero. Any potential factor must be a divisor of left and right: 1 < factor <= gcd(left, right)
When looking at all 10-pandigital numbers (there are only 10! = 3628800 such numbers), I observed that all factor are multiples of 3.
Modifications by HackerRank
The program is forced to use a certain output format (no big problem).
My program fails to process the massive amount of test cases.
Note
Chances are that splitting the pandigital number into more than two parts produces valid results, too.
Maybe I just got lucky by finding the correct solution.
Interactive test
You can submit your own input to my program and it will be instantly processed at my server:
This is equivalent toecho "1 1987654320" | ./170
Output:
Note: the original problem's input 9876543210
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 <string>
#include <iostream>
#include <algorithm>
// true, if x contains only distinct digits
bool isPandigital(unsigned long long x)
{
unsigned char used[10] = { 0 };
while (x > 0)
{
auto digit = x % 10;
// digit already used ?
if (used[digit] == 1)
return false;
used[digit]++;
x /= 10;
}
return true;
}
// greatest common divisor
unsigned int gcd(unsigned int a, unsigned int b)
{
while (a != 0)
{
unsigned int c = a;
a = b % a;
b = c;
}
return b;
}
int main()
{
unsigned int tests = 1;
std::cin >> tests;
while (tests--)
{
std::string current = "9876543210";
std::cin >> current;
// find next smaller pandigital number (only to avoid malicious input)
unsigned long long adjusted = std::stoll(current);
if (adjusted < 1023456789)
adjusted = 1023456789;
while (!isPandigital(adjusted))
adjusted--;
current = std::to_string(adjusted);
// start search
bool found = false;
do
{
// split into two parts and check each common divisor
for (size_t split = 1; split < current.size() && !found; split++)
{
// must not begin with a zero
if (current[0] == '0' || current[split] == '0')
continue;
auto left = std::stoll(current.substr(0, split));
auto right = std::stoll(current.substr(split));
// any common divisors ?
unsigned int shared = gcd(left, right);
const unsigned int MultipleOfThree = 3; // I saw that all divisors are always multiples of three
for (unsigned int factor = MultipleOfThree; factor <= shared; factor += MultipleOfThree)
{
// analyze all common divisors
if (left % factor == 0 &&
right % factor == 0)
{
// combine all digits
unsigned int one = left / factor;
unsigned int two = right / factor;
std::string sequence = std::to_string(factor) +
std::to_string(one) +
std::to_string(two);
// must have exactly 10 pandigital digits
if (sequence.size() == 10 && isPandigital(std::stoll(sequence)))
{
found = true;
std::cout << factor << "*(" << one << "," << two << ")=" << current << std::endl;
break;
}
}
}
}
// done ?
if (found)
break;
} while (std::prev_permutation(current.begin(), current.end()));
}
return 0;
}
This solution contains 11 empty lines, 12 comments and 3 preprocessor commands.
Benchmark
The correct solution to the original Project Euler problem was found in 0.08 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
June 19, 2017 submitted solution
June 19, 2017 added comments
Hackerrank
see https://www.hackerrank.com/contests/projecteuler/challenges/euler170
My code solves 2 out of 21 test cases (score: 5%)
I failed 0 test cases due to wrong answers and 19 because of timeouts
Difficulty
Project Euler ranks this problem at 70% (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=170 - the best forum on the subject (note: you have to submit the correct solution first)
Code in various languages:
Python github.com/smacke/project-euler/blob/master/python/170.py (written by Stephen Macke)
C++ github.com/HaochenLiu/My-Project-Euler/blob/master/170.cpp (written by Haochen Liu)
C++ github.com/Meng-Gen/ProjectEuler/blob/master/170.cc (written by Meng-Gen Tsai)
C++ github.com/roosephu/project-euler/blob/master/170.cpp (written by Yuping Luo)
Java github.com/thrap/project-euler/blob/master/src/Java/Problem170.java (written by Magnus Solheim Thrap)
Perl github.com/shlomif/project-euler/blob/master/project-euler/170/euler-170.pl (written by Shlomi Fish)
Sage github.com/roosephu/project-euler/blob/master/170.sage (written by Yuping Luo)
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 |
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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 169 - Exploring the number of different ways a number ... | Finding numbers for which the sum of the squares ... - problem 171 >> |