<< problem 109 - Darts | Bouncy numbers - problem 112 >> |

# Problem 111: Primes with runs

(see projecteuler.net/problem=111)

Considering 4-digit primes containing repeated digits it is clear that they cannot all be the same:

1111 is divisible by 11, 2222 is divisible by 22, and so on.

But there are nine 4-digit primes containing three ones:

1117, 1151, 1171, 1181, 1511, 1811, 2111, 4111, 8111

We shall say that M(n, d) represents the maximum number of repeated digits for an n-digit prime where d is the repeated digit,

N(n, d) represents the number of such primes, and S(n, d) represents the sum of these primes.

So M(4, 1) = 3 is the maximum number of repeated digits for a 4-digit prime where one is the repeated digit,

there are N(4, 1) = 9 such primes, and the sum of these primes is S(4, 1) = 22275.

It turns out that for d = 0, it is only possible to have M(4, 0) = 2 repeated digits, but there are N(4, 0) = 13 such cases.

In the same way we obtain the following results for 4-digit primes.

Digit dM(4,d)N(4,d)S(4,d)

021367061

13922275

2312221

331246214

4328888

5315557

6316661

73957863

8318887

93748073

For d = 0 to 9, the sum of all S(4, d) is 273700.

Find the sum of all S(10, d).

# Algorithm

`search`

iterates through all numbers and calls `isPrime`

to figure out whether it's a prime number.

There are a few gotchas:

1. While iterating, I create all possible strings. They may contain leading zeros, which are not allowed.

2. Add numbers into the sequence of repeated digits only if the number's digits are in non-decreasing order: 125 is okay, 122 is okay, 121 isn't.

## Modifications by HackerRank

I should analyze numbers with up to 40 digits (instead of 10). My code is too slow to handle anything with more than 12 digits.

Even worse, my routines can't process numbers exceeding 64 bits (which is about 19 digits).

# 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 <iostream>
#include <string>
#include <vector>
#include <algorithm>
#define ORIGINAL
// return true if x is prime, based on trial division (not suitable for large numbers)

bool isPrime(unsigned long long x)
{
// trial divison by 2
if (x % 2 == 0)
return x == 2;
// trial division by odd numbers
for (unsigned long long i = 3; i*i <= x; i++)
if (x % i == 0)
return false;
return true;
}
// find all numbers where "digit" is repeated "repeat" times and "extraDigits" digits are added
// e.g. 10007 is found by search(0, 3, 2)

unsigned long long search(unsigned int digit, unsigned int repeat, unsigned int extraDigits, bool printPrimes = false)
{
// sum of all matching primes
unsigned long long sum = 0;
std::vector<unsigned long long> matches; // Hackerrank only: print matches in ascending order
// create string with a digit repeated a few times
std::string sameDigit(repeat, digit + '0');
// insert numbers i where 0 <= i < maxExtra
// e.g. if extraDigits = 3 then from 0 to 999 (actually from "000" to "999")
unsigned long long maxExtra = 1;
for (unsigned int i = 1; i <= extraDigits; i++)
maxExtra *= 10;
// let's insert all those numbers !
for (unsigned long long extra = 0; extra < maxExtra; extra++)
{
// convert to string
auto current = std::to_string(extra);
// is in ascending order ?
auto sorted = current;
std::sort(sorted.begin(), sorted.end());
if (current != sorted)
continue;
// add zeros if "extra" has too few digits
while (current.size() < extraDigits)
current = '0' + current;
// and add the bunch of identical digits
current += sameDigit;
// sort everything to ensure that std::next_permutation works properly
std::sort(current.begin(), current.end());
do
{
// no leading zeros
if (current.front() == '0')
continue;
// no even numbers
if (current.back() % 2 == 0) // ASCII code of even digits is even, too
continue;
// is it a prime ?
unsigned long long num = std::stoll(current);
if (isPrime(num))
{
sum += num;
if (printPrimes)
matches.push_back(num);
}
} while (std::next_permutation(current.begin(), current.end()));
}
// Hackerrank only: print all primes we found
if (printPrimes && !matches.empty())
{
std::sort(matches.begin(), matches.end());
for (auto x : matches)
std::cout << x << " ";
}
return sum;
}
int main()
{
#ifdef ORIGINAL
// total number of digits
unsigned int digits = 10;
std::cin >> digits;
// sum of all matching primes
unsigned long long sum = 0;
// iterate over all repeated digits
for (unsigned int i = 0; i <= 9; i++)
{
// try to use as many repeated digits as possible
for (unsigned int repeated = digits - 1; repeated >= 1; repeated--)
{
auto insertDigits = digits - repeated;
auto found = search(i, repeated, insertDigits);
// found at least one prime ?
if (found > 0)
{
sum += found;
break;
}
}
}
std::cout << sum << std::endl;
#else
unsigned int tests;
std::cin >> tests;
while (tests--)
{
unsigned int digits;
unsigned int sameDigit;
std::cin >> digits >> sameDigit;
if (digits < 19) // my code can't handle > 64 bits at the moment
{
// iterate over all repeated digits
for (unsigned int repeated = digits - 1; repeated >= 1; repeated--)
{
auto insertDigits = digits - repeated;
if (search(sameDigit, repeated, insertDigits, true) > 0)
break;
}
}
std::cout << std::endl;
}
#endif
return 0;
}

This solution contains 23 empty lines, 25 comments and 8 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 4 | ./111`

Output:

*Note:* the original problem's input `10`

__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 **0.08** seconds on a Intel® Core™ i7-2600K CPU @ 3.40GHz.

(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

May 17, 2017 submitted solution

May 17, 2017 added comments

# Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler111

My code solved **1** out of **20** test cases (score: **100%**)

I failed **0** test cases due to wrong answers and **19** because of timeouts

# Difficulty

Project Euler ranks this problem at **45%** (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=111 - **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-111-10-digit-primes-repeated-digits/ (written by Kristian Edlund)

Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p111.java (written by Nayuki)

# 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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