<< 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.

# 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 never 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.

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

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My username at Project Euler is

**stephanbrumme**while it's stbrumme at Hackerrank.

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