<< problem 48 - Self powers | Consecutive prime sum - problem 50 >> |
Problem 49: Prime permutations
(see projecteuler.net/problem=49)
The arithmetic sequence, 1487, 4817, 8147, in which each of the terms increases by 3330, is unusual in two ways:
(i) each of the three terms are prime, and,
(ii) each of the 4-digit numbers are permutations of one another.
There are no arithmetic sequences made up of three 1-, 2-, or 3-digit primes, exhibiting this property, but there is one other 4-digit increasing sequence.
What 12-digit number do you form by concatenating the three terms in this sequence?
My Algorithm
My function fingerprint
counts how often each digit occurs and produces an integer (which may have up to 10 digits).
The n-th decimal digit of the result represents how often the digit n occurs in the input, e.g.
fingerprint(454430) = 131001
because 5
appears once, 4
three times, 3
once, no 2
, no 1
and a single zero.
fingerprint
has the nice property that two number with the same fingerprint are a permutation of each other.
After generating all primes
, their fingerprints
are stored.
All permutations of a prime number, which are prime themselves, will be added to a list of candidates
.
There must be at least sequenceLength
(it's 3
in the original problem) candidates.
However, some candidates may not have the proper distance to each other: that's why I compute the differences
of each candidate prime to each other.
Only if at least sequenceLength
primes share the same distance to at least one other prime, then we may have a result.
Unfortunately, pairs may have the same distance diff = |p_a - p_b| = |p_c - p_d| but are disjunct: diff != |p_a - p_c|.
For example, the primes 3, 5, 17, 19 have a pair-wise distance of 2 (3-5 and 17-19) but there is no way to connect 3 and 5 to 17 and 19.
Therefore the program tries to start at every candidate prime p_i and looks for the longest sequence p_i + diff, p_i + 2 * diff, p_i + 3 * diff, ...
where each element p_i + k * diff is part of the candidates.
If such a sequence was found, the program repeats the same process but connects all elements to a long string.
All strings are stored in an std::set
which is automatically ordered.
Modifications by HackerRank
Substantial parts of my code are due to Hackerrank's modifications: the sequenceLength
may be 3 or 4 and a user-defined upper limit
exists.
Default values for the original problem would be 10000
and 3
.
Interactive test
You can submit your own input to my program and it will be instantly processed at my server:
This live test is based on the Hackerrank problem.
This is equivalent toecho "2000 3" | ./49
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.
The code contains #ifdef
s to switch between the original problem and the Hackerrank version.
Enable #ifdef ORIGINAL
to produce the result for the original problem (default setting for most problems).
#include <set>
#include <map>
#include <iostream>
#include <string>
#include <algorithm>
// count how often each digit appears: result's n-th digit describes how often n appears in x
// e.g. 454430 => 131001
// because 5 appears once, 4 three times, 3 once, no 2, no 1 and a single zero
unsigned long long fingerprint(unsigned int x)
{
unsigned long long result = 0;
while (x > 0)
{
auto digit = x % 10;
x /= 10;
unsigned long long pos = 1;
for (unsigned int i = 1; i <= digit; i++)
pos *= 10;
result += pos;
}
return result;
}
int main()
{
unsigned int limit = 10000;
unsigned int sequenceLength = 4;
std::cin >> limit >> sequenceLength;
// find primes (simple sieve)
std::set<unsigned int> primes;
primes.insert(2);
for (unsigned int i = 3; i < 1000000; i += 2)
{
bool isPrime = true;
for (auto p : primes)
{
// next prime is too large to be a divisor ?
if (p*p > i)
break;
// divisible ? => not prime
if (i % p == 0)
{
isPrime = false;
break;
}
}
// yes, we have a prime number
if (isPrime)
primes.insert(i);
}
// count fingerprints of all prime numbers
std::map<unsigned long long, unsigned int> fingerprints;
for (auto p : primes)
fingerprints[fingerprint(p)]++;
// [length] => [merged primes, alphabetically ordered]
std::map<unsigned int, std::set<std::string>> result;
// iterate through all primes
for (auto p : primes)
{
// at least three digits ...
if (p < 1000)
continue;
// but not too far ...
if (p >= limit)
break;
// too few primes sharing this fingerprint ?
if (fingerprints[fingerprint(p)] < 3)
continue;
// generate all digit permutations
std::string digits = std::to_string(p);
std::sort(digits.begin(), digits.end());
// find all permutations which are primes
std::set<unsigned int> candidates;
do
{
// first digit can't be zero
if (digits[0] == '0')
continue;
// convert to an integer
unsigned int permuted = std::stoi(digits);
// permutation must be prime, too
if (primes.count(permuted) == 0)
continue;
// we already had this sequence ?
if (permuted < p)
break;
// yes, a valid prime
candidates.insert(permuted);
} while (std::next_permutation(digits.begin(), digits.end()));
// too few candidates ?
if (candidates.size() < sequenceLength)
continue;
// compute differences of each prime to each other prime
// [difference] => [primes that are that far away from another prime]
std::map<unsigned int, std::set<unsigned int>> differences;
for (auto bigger : candidates)
for (auto smaller : candidates)
{
// ensure smaller < bigger
if (smaller >= bigger)
break;
// store both primes
differences[bigger - smaller].insert(bigger);
differences[bigger - smaller].insert(smaller);
}
// walk through all differences
for (auto d : differences)
{
// at least 3 or 4 numbers must be involved in a sequence
if (d.second.size() < sequenceLength)
continue;
// current difference
auto diff = d.first;
// potential numbers for a sequence
auto all = d.second;
// could be a false alarm if disjunct pairs have the same difference
// we need a sequence ...
for (auto start : all)
{
// out of bounds ?
if (start >= limit)
continue;
// count numbers which can be reached by repeatedly adding our current difference
unsigned int followers = 0;
unsigned int next = start + diff;
while (all.count(next) != 0)
{
followers++;
next += diff;
}
// found enough ? => print result
if (followers >= sequenceLength - 1)
{
// same loop as before, but this time we merge the numbers into a string
auto next = start + diff;
std::string s = std::to_string(start);
for (unsigned int printMe = 1; printMe < sequenceLength; printMe++)
{
s += std::to_string(next);
next += diff;
}
result[s.size()].insert(s);
}
}
}
}
//#define ORIGINAL
// print everything, ordered by length and content
for (auto length : result)
for (auto x : length.second)
#ifdef ORIGINAL
if (x != "148748178147") // skip that solution
#endif
std::cout << x << std::endl;
return 0;
}
This solution contains 25 empty lines, 37 comments and 7 preprocessor commands.
Benchmark
The correct solution to the original Project Euler problem was found in 0.17 seconds on an Intel® Core™ i7-2600K CPU @ 3.40GHz.
Peak memory usage was about 6 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
February 27, 2017 submitted solution
April 20, 2017 added comments
Hackerrank
see https://www.hackerrank.com/contests/projecteuler/challenges/euler049
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 hard.
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.
Similar problems at Project Euler
Problem 52: Permuted multiples
Note: I'm not even close to solving all problems at Project Euler. Chances are that similar problems do exist and I just haven't looked at them.
Links
projecteuler.net/thread=49 - 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-49-arithmetic-sequences-primes-permutations/ (written by Kristian Edlund)
C github.com/eagletmt/project-euler-c/blob/master/40-49/problem49.cc (written by eagletmt)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p049.java (written by Nayuki)
Javascript github.com/dsernst/ProjectEuler/blob/master/49 Prime permutations.js (written by David Ernst)
Go github.com/frrad/project-euler/blob/master/golang/Problem049.go (written by Frederick Robinson)
Mathematica github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p049.mathematica (written by Nayuki)
Scala github.com/samskivert/euler-scala/blob/master/Euler049.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 | |
[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 48 - Self powers | Consecutive prime sum - problem 50 >> |