Problem 37: Truncatable primes

(see projecteuler.net/problem=37)

The number 3797 has an interesting property. Being prime itself, it is possible to continuously remove digits from left to right,
and remain prime at each stage: 3797, 797, 97, and 7. Similarly we can work from right to left: 3797, 379, 37, and 3.

Find the sum of the only eleven primes that are both truncatable from left to right and right to left.

NOTE: 2, 3, 5, and 7 are not considered to be truncatable primes.

Algorithm

A generic prime sieve is mixed with my check for "prime-truncatability": as soon as a prime number is found, I remove digits step-by-step from the right side
and make sure that the truncated number is still a prime number. In pseudo-code:

while (number > 0 && primes.count(number) != 0)
remove_one_digit(number) ← see explanation below

If number == 0 at the end of the while-loop then number is truncatable and prime.

Removing the right-most digit is simple: just divide by 10 and ignore any remainder. See the code that processes right.
Removing the left-most digit takes a little more effort (look out for left and factor):
Find that largest power-of-10 that is still smaller than the current number (that's my factor).
Then the remainder of left % factor chops off the left-most digit.

Modifications by HackerRank

Hackerrank gave indirectly away that all such numbers are less than 1000000 (which is confirmed by OEIS A020994).
Hackerrank's problem asks for a user-defined upper limit.

My code

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

#include <set>
#include <iostream>
 
int main()
{
// find all primes up to
unsigned int n;
std::cin >> n; // 1000000 is sufficient for the original problem
 
// will contain all primes found so far
std::set<unsigned int> primes;
// all single-digit prime numbers (2,3,5,7) are not truncatable by definition
primes.insert(2);
primes.insert(3);
primes.insert(5);
primes.insert(7);
 
unsigned int sum = 0;
// check prime numbers with at least two digits
// note: even numbers cannot be prime (except 2)
for (unsigned int i = 11; i < n; i += 2)
{
bool isPrime = true;
// check against all known primes
for (auto p : primes)
{
// no more prime factors possible
if (p*p > i)
break;
 
// divisible by another prime ? => i is not prime
if (i % p == 0)
{
isPrime = false;
break;
}
}
 
if (!isPrime)
continue;
 
// now we have a prime
primes.insert(i);
 
// check whether truncable from the right side
auto right = i;
// remove right-most digit if still prime and until no digits remain
while (right > 0 && primes.count(right) != 0)
right /= 10;
// pass only if all digits were successfully removed
if (right != 0)
continue;
 
// same idea from the left side
auto left = i;
// find position of left-most digit
unsigned int factor = 1;
while (factor * 10 <= left)
factor *= 10;
// remove left-most digit if still prime and until no digits remain
while (left > 0 && primes.count(left) != 0)
{
// fast version:
left %= factor;
// slower version: subtract until highest digit is completely gone (=zero)
//while (left >= factor)
// left -= factor;
 
// okay, next digit is 10 times smaller
factor /= 10;
}
// pass only if all digits were successfully removed
if (left != 0)
continue;
 
// yeah, passed all tests !
sum += i;
}
 
std::cout << sum << std::endl;
return 0;
}

This solution contains 11 empty lines, 22 comments and 2 preprocessor commands.

Interactive test

You can submit your own input to my program and it will be instantly processed at my server:

Input data (separated by spaces or newlines):

This is equivalent to
echo 10000 | ./37

Output:

(please click 'Go !')

Note: the original problem's input 1000000 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)

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 23, 2017 submitted solution
April 12, 2017 added comments

Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler037

My code solves 4 out of 4 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=37 - 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-37-truncatable-primes/ (written by Kristian Edlund)
Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p037.java (written by Nayuki)
Mathematica: github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p037.mathematica (written by Nayuki)
C: github.com/eagletmt/project-euler-c/blob/master/30-39/problem37.c (written by eagletmt)
Go: github.com/frrad/project-euler/blob/master/golang/Problem037.go (written by Frederick Robinson)
Javascript: github.com/dsernst/ProjectEuler/blob/master/37 Truncatable primes.js (written by David Ernst)
Scala: github.com/samskivert/euler-scala/blob/master/Euler037.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.
red problems are solved but exceed the time limit of one minute.

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The 206 solved problems had an average difficulty of 27.5% at Project Euler and
I scored 12,626 points (out of 14300 possible points, top rank was 20 out ouf ≈60000 in July 2017) at Hackerrank's Project Euler+.
Look at my progress and performance pages to get more details.

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.
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