<< problem 61 - Cyclical figurate numbers | Powerful digit counts - problem 63 >> |
Problem 62: Cubic permutations
(see projecteuler.net/problem=62)
The cube, 41063625 (345^3), can be permuted to produce two other cubes: 56623104 (384^3) and 66430125 (405^3).
In fact, 41063625 is the smallest cube which has exactly three permutations of its digits which are also cube.
Find the smallest cube for which exactly five permutations of its digits are cube.
My Algorithm
A function fingerprint
was already used in problem 49 and problem 52: if fingerprint(a) == fingerprint(b)
then a
is a permutation of b
.
Step 1 of my program computes all cubes from 1^3 to maxCube^3and stores them indexed by their fingerprint (see matches
).
Step 2 goes through all fingerprints with the specified number of permutations and transfers each fingerprint's first cube (the "smallest") to an std::set
called smallest
.
Finally, smallest^3
is printed - using the fact that each std::set
is automatically sorted (needed for the Hackerrank version of the problem).
Modifications by HackerRank
The modified problem requires all permutations to be below maxCube^3.
When running the problem with the official solution (parameter xyz 5
where xyz
is the solution) then it will fail to find the correct solution.
The reason is that all but the initial permutations of maxCube^3 are bigger than maxCube^3.
Therefore the solution is only found if all cubes are processed which produce the solution and all bigger cubes that have still the same number of digits.
To be safe, the parameter should be solution * {^3}sqrt{10} approx solution * 2.16.
Interactive test
You can submit your own input to my program and it will be instantly processed at my server:
This is equivalent toecho "1000 3" | ./62
Output:
(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 <map>
#include <set>
#include <vector>
#include <iostream>
// count how often each digit occurs,
// store zero at lowest position, then ones, etc.
// e.g. 5063 means 3 zeros, 6 ones, no twos, 5 threes and nothing else
// note: can't handle input values with 2^6 or more identical digits
unsigned long long fingerprint(unsigned long long x)
{
unsigned long long result = 0;
while (x > 0)
{
// extract lowest digit
auto digit = x % 10;
x /= 10;
// subdivide 64 bit integer into 10 "digit counters", each 6 bits wide
// => each digit may occur up to 2^6=64 times, more than enough ...
const auto BitsPerDigit = 6;
result += 1ULL << (BitsPerDigit * digit);
}
return result;
}
int main()
{
unsigned int maxCube = 10000;
unsigned int numPermutations = 5;
std::cin >> maxCube >> numPermutations;
// [fingerprint] => [list of numbers, where number^3 produced that fingerprint]
std::map<unsigned long long, std::vector<unsigned int>> matches;
for (unsigned int i = 1; i < maxCube; i++)
{
// find fingerprint
auto cube = (unsigned long long)i * i * i;
// add current number to the fingerprint's list
matches[fingerprint(cube)].push_back(i);
}
// extract all smallest cube, std::set is sorting them
std::set<unsigned long long> smallest;
for (auto m : matches)
// right number of permutations ?
if (m.second.size() == numPermutations)
smallest.insert(m.second.front());
// print in ascending order
for (auto s : smallest)
std::cout << (s*s*s) << std::endl;
return 0;
}
This solution contains 7 empty lines, 13 comments and 4 preprocessor commands.
Benchmark
The correct solution to the original Project Euler problem was found in less than 0.01 seconds on an Intel® Core™ i7-2600K CPU @ 3.40GHz.
Peak memory usage was about 3 MByte.
(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
March 1, 2017 submitted solution
April 22, 2017 added comments
Hackerrank
see https://www.hackerrank.com/contests/projecteuler/challenges/euler062
My code solves 10 out of 10 test cases (score: 100%)
Difficulty
Project Euler ranks this problem at 15% (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=62 - 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-62-cube-five-permutations/ (written by Kristian Edlund)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p062.java (written by Nayuki)
Go github.com/frrad/project-euler/blob/master/golang/Problem062.go (written by Frederick Robinson)
Scala github.com/samskivert/euler-scala/blob/master/Euler062.scala (written by Michael Bayne)
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 | |
the flashing problem is the one I solved most recently |
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I scored 13,486 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 61 - Cyclical figurate numbers | Powerful digit counts - problem 63 >> |