<< problem 36 - Double-base palindromes | Pandigital multiples - problem 38 >> |

# 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.

# My 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.

# Interactive test

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

This is equivalent to`echo 10000 | ./37`

Output:

*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)*

# 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.

# 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:

C#: 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)

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.

# 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 |

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I scored 13,386 points (out of 15600 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 36 - Double-base palindromes | Pandigital multiples - problem 38 >> |