<< problem 51 - Prime digit replacements | Combinatoric selections - problem 53 >> |

# Problem 52: Permuted multiples

(see projecteuler.net/problem=52)

It can be seen that the number, 125874, and its double, 251748, contain exactly the same digits, but in a different order.

Find the smallest positive integer, x, such that 2x, 3x, 4x, 5x, and 6x, contain the same digits.

# My Algorithm

My function `fingerprint`

counts how often each digit occurs and produces an integer (which may have up to 10 digits).

The n-th decimal digit of the result represents how often the digit n occurs in the input, e.g.

`fingerprint(454430) = 131001`

because `5`

appears once, `4`

three times, `3`

once, no `2`

, no `1`

and a single zero.

`fingerprint`

has the nice property that two number with the same fingerprint are a permutation of each other

(phrased in the words of the problem statement: "contain the same digits").

*Note:* my fingerprint technique allows only up 9 identical digits which is okay because `x`

has at most seven digits.

I compute the fingerprint of each number `i`

, beginning with 1, and multiply it by 2, 3, 4, ...

If the product still has the same fingerprint, then it is a permutation.

## Modifications by HackerRank

The number of multiples can be adjusted from 2 to 6 (the latter being the default value for the original problem).

## Note

The is plenty of room for optimization. For example, if `maxMultiple >= 5`

then the first digit of `i`

must be a `1`

.

# 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>
// I generate a "fingerprint" for each number:
// e.g. a fingerprint of 40231 means that the parameter had
// 1 zero
// 3 ones
// 2 threes
// no fours
// 4 fives
// and no sixes, sevens, ...

unsigned long long fingerprint(unsigned int x)
{
unsigned long long result = 0;
while (x > 0)
{
// extract right-most digit
auto digit = x % 10;
x /= 10;
// add 10^digit
unsigned long long pos = 1;
for (unsigned int i = 1; i <= digit; i++)
pos *= 10;
result += pos;
}
return result;
}
int main()
{
// the result can be found with 1000000 6
unsigned int maxNumber = 1000000;
unsigned int maxMultiple = 6;;
std::cin >> maxNumber >> maxMultiple;
// look at all numbers
for (unsigned int i = 1; i <= maxNumber; i++)
{
// initial fingerprint
auto id = fingerprint(i);
bool found = true;
for (unsigned int multiple = 2; multiple <= maxMultiple; multiple++)
// mismatch ? => abort
if (id != fingerprint(i * multiple))
{
found = false;
break;
}
// print result
if (found)
{
//#define ORIGINAL

#ifdef ORIGINAL
std::cout << i << std::endl;
return 0;
#endif
for (unsigned int multiple = 1; multiple <= maxMultiple; multiple++)
std::cout << (i * multiple) << " ";
std::cout << std::endl;
}
}
return 0;
}

This solution contains 10 empty lines, 16 comments and 3 preprocessor commands.

# Interactive test

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

This live test is based on the Hackerrank problem.

This is equivalent to`echo "125875 2" | ./52`

Output:

*(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 0.02 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

February 27, 2017 submitted solution

April 20, 2017 added comments

# Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler052

My code solves **10** out of **10** 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.

# Similar problems at Project Euler

Problem 49: Prime permutations

*Note:* I'm not even close to solving all problems at Project Euler. Chances are that similar problems do exist and I just haven't looked at them.

# Links

projecteuler.net/thread=52 - **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-52-integer-same-digits/ (written by Kristian Edlund)

Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p052.java (written by Nayuki)

Mathematica: github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p052.mathematica (written by Nayuki)

Go: github.com/frrad/project-euler/blob/master/golang/Problem052.go (written by Frederick Robinson)

Javascript: github.com/dsernst/ProjectEuler/blob/master/52 Permuted multiples.js (written by David Ernst)

Scala: github.com/samskivert/euler-scala/blob/master/Euler052.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 or the memory limit of 256 MByte.

*Please click on a problem's number to open my solution to that problem:*

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I scored 12,983 points (out of 15100 possible points, top rank was 17 out of ≈60000 in August 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.

# 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 51 - Prime digit replacements | Combinatoric selections - problem 53 >> |