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?

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.

My code

… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too.

The code contains #ifdefs 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.

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.

Input data (separated by spaces or newlines):

This is equivalent to
echo "2000 3" | ./49

Output:

(please click 'Go !')

(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 0.17 seconds on a 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=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 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 never 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:

Python: www.mathblog.dk/project-euler-49-arithmetic-sequences-primes-permutations/ (written by Kristian Edlund)
Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p049.java (written by Nayuki)
Mathematica: github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p049.mathematica (written by Nayuki)
C: github.com/eagletmt/project-euler-c/blob/master/40-49/problem49.cc (written by eagletmt)
Go: github.com/frrad/project-euler/blob/master/golang/Problem049.go (written by Frederick Robinson)
Javascript: github.com/dsernst/ProjectEuler/blob/master/49 Prime permutations.js (written by David Ernst)
Scala: github.com/samskivert/euler-scala/blob/master/Euler049.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 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
201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225
226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250
The 163 solved problems had an average difficulty of 22.2% at Project Euler and I scored 11,907 points (out of 13200) at Hackerrank's Project Euler+.
My username at Project Euler is stephanbrumme while it's stbrumme at Hackerrank.
more about me can be found on my homepage, especially in my coding blog.
some names mentioned on this site may be trademarks of their respective owners.
thanks to the KaTeX team for their great typesetting library !