<< problem 37 - Truncatable primes | Integer right triangles - problem 39 >> |

# Problem 38: Pandigital multiples

(see projecteuler.net/problem=38)

Take the number 192 and multiply it by each of 1, 2, and 3:

192 * 1 = 192

192 * 2 = 384

192 * 3 = 576

By concatenating each product we get the 1 to 9 pandigital, 192384576. We will call 192384576 the concatenated product of 192 and (1,2,3)

The same can be achieved by starting with 9 and multiplying by 1, 2, 3, 4, and 5, giving the pandigital, 918273645, which is the concatenated product of 9 and (1,2,3,4,5).

What is the largest 1 to 9 pandigital 9-digit number that can be formed as the concatenated product of an integer with (1,2, ... , n) where n > 1?

# Algorithm

A bitmask tracks whether a number is pandigital (or not): if the n-th bit of `bitsUsed`

is 1 then n is a digit of `pandigital`

.

A truly pandigital number has a bitmask of `bitsAll = (1 << 9) | (1 << 8) | ... | (1 << 2) | (1 << 1)`

.

According to the problem statement, a digit must not be zero. That means the lowest bit is always 0:

bitsAll_9=1022 (and bitsAll_8=510 for alternate Hackerrank input).

My program process all numbers `i`

from 2 to `maxFactor`

and multiplies them with 1, 2, ... until the concatenated result's `bitsUsed == bitsAll`

.

Each `product = i * multiplier`

(where `multiplier = 1, 2, ...`

) is split into its digits and each digit added to the bitmask `bitsUsed`

.

When a collision occurs (that digit was already used before), then `bitsUsed`

is set to an invalid value and the next number `i`

can be processed.

## Modifications by HackerRank

The result may have either 8 or 9 digits and there is an upper limit for the start value.

## Note

`maxFactor`

cannot exceed 10000 because then `i*1`

+`i*2`

will have more than 9 digits.

A few things are probably much easier when converting all numbers to `std::string`

- at the cost of speed.

# My code

… was written in C++ 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>
int main()
{
unsigned int maxFactor, maxDigit; // 10000 and 9 for the original problem
std::cin >> maxFactor >> maxDigit;
// bitmask if all digits are used
unsigned int bitsAll = 0;
for (unsigned int i = 1; i <= maxDigit; i++)
bitsAll |= 1 << i;
#define ORIGINAL
#ifdef ORIGINAL
// largest pandigital number found so far
unsigned int largest = 0;
#endif
// try all numbers
for (unsigned int i = 2; i <= maxFactor; i++)
{
// the whole pandigital number
unsigned int pandigital = 0;
// multiply i by 1,2,3,...
unsigned int multiplier = 1;
// bitmask of used digits (nth bit is set if pandigital contains digit n)
unsigned int bitsUsed = 0;
while (bitsUsed < bitsAll)
{
// next step
unsigned int product = i * multiplier;
// extract right-most digit
while (product > 0)
{
// extract right-most digit
unsigned int digit = product % 10;
// remove it
product /= 10;
// make room to add i*multiplier lateron
pandigital *= 10;
// remember all digits we used
unsigned int bitMask = 1 << digit;
// we already had this digit ?
if (digit == 0 || (bitsUsed & bitMask) != 0)
{
bitsUsed = bitsAll + 1; // set to an invalid value
break;
}
// mark current digit as "used"
bitsUsed |= bitMask;
}
// keep going in the sequence
pandigital += i * multiplier;
multiplier++;
}
// enough digits generated ?
if (bitsUsed == bitsAll)
{
#ifdef ORIGINAL
if (largest < pandigital)
largest = pandigital;
#else
std::cout << i << std::endl;
#endif
}
}
#ifdef ORIGINAL
std::cout << largest << std::endl;
#endif
return 0;
}

This solution contains 15 empty lines, 16 comments and 9 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 "200 9" | ./38`

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 **less than 0.01** 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 25, 2017 submitted solution

April 12, 2017 added comments

# Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler038

My code solved **6** out of **6** 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=38 - **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-38-pandigital-multiplying-fixed-number/ (written by Kristian Edlund)

Haskell: github.com/nayuki/Project-Euler-solutions/blob/master/haskell/p038.hs (written by Nayuki)

Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p038.java (written by Nayuki)

Mathematica: github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p038.mathematica (written by Nayuki)

C: github.com/eagletmt/project-euler-c/blob/master/30-39/problem38.c (written by eagletmt)

Go: github.com/frrad/project-euler/blob/master/golang/Problem038.go (written by Frederick Robinson)

Javascript: github.com/dsernst/ProjectEuler/blob/master/38 Pandigital multiples.js (written by David Ernst)

Scala: github.com/samskivert/euler-scala/blob/master/Euler038.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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<< problem 37 - Truncatable primes | Integer right triangles - problem 39 >> |