<< 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.

# 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:

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)*

# 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 solved **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 never 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 already 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:*

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