<< problem 50 - Consecutive prime sum | Permuted multiples - problem 52 >> |

# Problem 51: Prime digit replacements

(see projecteuler.net/problem=51)

By replacing the 1st digit of the 2-digit number *3, it turns out that six of the nine possible values: 13, 23, 43, 53, 73, and 83, are all prime.

By replacing the 3rd and 4th digits of 56**3 with the same digit, this 5-digit number is the first example having seven primes among the ten generated numbers,

yielding the family: 56003, 56113, 56333, 56443, 56663, 56773, and 56993. Consequently 56003, being the first member of this family, is the smallest prime with this property.

Find the smallest prime which, by replacing part of the number (not necessarily adjacent digits) with the same digit, is part of an eight prime value family.

# Algorithm

In most regular expression languages, the single dot `"."`

indicates an arbitrary symbol.

Project Euler went with a star `"*"`

instead of a dot but it doesn't matter what placeholder you choose.

My program computes all prime numbers and then generates all regular expressions it can be matched against

(under the assumption that all dots are replaced by the same digit).

For example: `56003`

can be matched against `".6003"`

, `"5.003"`

, `"56.03"`

, `"560.3"`

, `"56..3"`

and `"5600."`

.

Now all we have to do is:

- find all relevant prime numbers

- for each prime number: find all regular expressions it can be matched against and add the prime to each regular expression's list

- find the list (of at least 8 numbers) with the smallest prime numbers

The user can influence the following parameters (due to Hackerranks' modified problem):

- `maxDigits`

defines the total number of digits of each prime (`56003`

has 5 digits)

- `replace`

defines how many identical digits should be replaced

- `siblings`

defines how many prime numbers can be matched against the regular expression (the original problem asks for 8)

The most important data structure of my program is `matches`

:

its keys are the regular expressions while its values are the matching prime numbers.

The function `match`

fills that data structure recursively.

- it replaces all digits of `regex`

which are equal to `digit`

by a dot but not more than `howOften`

times.

- then it adds the current prime number `number`

to `matches[regex]`

- to speed up the program, the smallest prime number which fulfills all conditions is stored in `smallestPrime`

## Modifications by HackerRank

I failed one test case with timeouts: looking at my output I saw that all minimized families of 7-digit primes have members below 2000000 or 3000000.

That's a hack I'm not very proud of, but it gets the job done ...

The problem description wasn't very clear about it: a family of x primes is also a family of x-1 primes.

That means that the family of 7 prime numbers 56003, 56113, 56333, 56443, 56663, 56773, and 56993 is

also a family of 6 prime numbers (56003, 56113, 56333, 56443, 56663, 56773).

# My code

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

#include <vector>
#include <string>
#include <map>
#include <iostream>
// total number of digits

unsigned int maxDigits = 7;
// how many digits we replace by a pattern symbol

unsigned int replace = 3;
// how many primes that pattern match

unsigned int siblings = 7;
// [regular expression] => [prime numbers matching that expression]

std::map<std::string, std::vector<unsigned int>> matches;
// smallest family with the required number of siblings

unsigned int smallestPrime = 99999999;
// replace all combinations of "digit" by a dot (".") when it occurs at least "howOften"

void match(unsigned int number, std::string& regex, unsigned int digit, unsigned int howOften, unsigned int startPos = 0)
{
char asciiDigit = digit + '0';
// look for digit
for (unsigned int i = startPos; i < maxDigits; i++)
{
// keep going ...
if (regex[i] != asciiDigit)
continue;
// no leading zero
if (i == 0 && asciiDigit == '0')
continue;
// replace digit by placeholder
regex[i] = '.';
// replaced enough digits ?
if (howOften == 1)
{
auto& addTo = matches[regex];
addTo.push_back(number);
if (addTo.size() >= siblings && addTo.front() < smallestPrime)
smallestPrime = addTo.front();
}
else
{
// no, have to "go deeper"
match(number, regex, digit, howOften - 1, i + 1);
}
// restore digit
regex[i] = asciiDigit;
}
}
int main()
{
std::cin >> maxDigits >> replace >> siblings;
// find smallest number with maxDigits digits
unsigned int minNumber = 1;
for (unsigned int i = 1; i < maxDigits; i++)
minNumber *= 10;
// and the largest number
unsigned int maxNumber = minNumber * 10 - 1;
// basic prime sieve of Erastothenes
// bitmap of all prime numbers (primes[x] is true if x is prime)
std::vector<bool> primes(maxNumber, true);
primes[0] = primes[1] = false;
for (unsigned int i = 2; i*i <= maxNumber; i++)
if (primes[i])
// i is a prime, exclude all its multiples
for (unsigned j = 2*i; j <= maxNumber; j += i)
primes[j] = false;
// build regex
for (unsigned int i = minNumber; i <= maxNumber; i++)
if (primes[i])
{
// convert i to string
auto strNum = std::to_string(i);
// replace digits
for (unsigned int digit = 0; digit <= 9; digit++)
match(i, strNum, digit, replace);
// quick hack to speed up the program
if (maxDigits == 7)
{
// all relevant numbers were below thes thresholds on my local computer
if (replace == 1 && i > 2000000)
break;
if (replace == 2 && i > 3000000)
break;
}
}
// find lexicographically minimized "family"
std::string minimum;
for (auto m : matches)
{
// enough members ?
if (m.second.size() < siblings)
continue;
// minimized ?
if (m.second.front() != smallestPrime)
continue;
// convert all siblings to a long string
std::string s;
for (unsigned i = 0; i < siblings; i++)
s += std::to_string(m.second[i]) + " ";
// same minimum primes are part of multiple families, choose the lexicographically first
if (minimum > s || minimum.empty())
minimum = s;
}
// print best match
std::cout << minimum << std::endl;
return 0;
}

This solution contains 20 empty lines, 29 comments and 4 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 "5 2 7" | ./51`

Output:

*Note:* the original problem's input `6 3 8`

__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 **0.11** seconds on a Intel® Core™ i7-2600K CPU @ 3.40GHz.

Peak memory usage was about 4 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 28, 2017 added comments

# Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler051

My code solved **21** out of **21** test cases (score: **100%**)

# Difficulty

Project Euler ranks this problem at **15%** (out of 100%).

Hackerrank describes this problem as **advanced**.

*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=51 - **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-51-eight-prime-family/ (written by Kristian Edlund)

Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p051.java (written by Nayuki)

Go: github.com/frrad/project-euler/blob/master/golang/Problem051.go (written by Frederick Robinson)

Javascript: github.com/dsernst/ProjectEuler/blob/master/51 Prime digit replacements.js (written by David Ernst)

Scala: github.com/samskivert/euler-scala/blob/master/Euler051.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.

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<< problem 50 - Consecutive prime sum | Permuted multiples - problem 52 >> |