Problem 4: Largest palindrome product

(see projecteuler.net/problem=4)

A palindromic number reads the same both ways.
The largest palindrome made from the product of two 2-digit numbers is 9009 = 91 * 99.
Find the largest palindrome made from the product of two 3-digit numbers.

Algorithm

The upper three digits of such a six-digit palindrome are a "mirrored" version of the lower three digits.
I wrote a function makePalindrome which takes a three-digit number and returns its six-digit palindrome.

There are only 899 six-digit palindromes because the upper three digits must be 100 ... 999
(no leading zeros allowed because then it wouldn't be a six-digit number anymore).

Beginning with 999, I loop "downwards" through all possible combinations trying to find a three-digit divisor.
A simple speedup is achieved by observing that at least one divisor i must be i>=100 and i^2<=palindrome.

Modifications by HackerRank

Hackerrank's problem asks for a variable maximum upper limit. It's 999999 for the original problem.

Note

I create six-digit palindromes by splitting a three-digit numbers into its single digits.
A possible alternative is to convert the three-digit number into a string and concatenate it with its reversed version.
However, often those string operations tend to be quite slow.

My code

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

#include <iostream>
 
// convert 3 digits to a 6 digit palindrome by mirroring and appending these 3 digits
// e.g. 234 becomes 234432
unsigned int makePalindrome(unsigned int x)
{
unsigned int result = x * 1000; // abc => abc000
result += x / 100; // a.. => a..00a
result += ((x / 10) % 10) * 10; // .b. => .b.0b.
result += (x % 10) * 100; // ..c => ..cc..
return result;
}
 
int main()
{
unsigned int tests;
std::cin >> tests;
while (tests--)
{
// Hackerrank has a variable upper limit (instead of 1000000)
unsigned int maximum;
std::cin >> maximum;
 
bool found = false;
// find all palindromes, beginning with the largest
// walk through all three-digit numbers
for (auto upper3 = maximum / 1000; upper3 >= 100 && !found; upper3--)
{
// "mirror" these three digits to create a six-digit palindrome
auto palindrome = makePalindrome(upper3);
// too big ?
if (palindrome >= maximum)
continue;
 
// split into two factors
for (unsigned int i = 100; i * i <= palindrome; i++)
if (palindrome % i == 0) // divisible ?
{
// make sure both factors must have three digits
auto other = palindrome / i;
if (other < 100 || other > 999)
continue;
 
std::cout << palindrome << std::endl;
found = true;
break;
}
}
}
return 0;
}

This solution contains 5 empty lines, 9 comments and 1 preprocessor command.

Interactive test

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

Number of test cases (1-5):

Input data (separated by spaces or newlines):

This is equivalent to
echo "1 888888" | ./4

Output:

(please click 'Go !')

Note: the original problem's input 1000000 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 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 23, 2017 submitted solution
March 27, 2017 added comments

Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler004

My code solved 4 out of 4 test cases (score: 100%)

Difficulty

Project Euler ranks this problem at 5% (out of 100%).

Hackerrank describes this problem as medium.

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=4 - 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-problem-4/ (written by Kristian Edlund)
Haskell: github.com/nayuki/Project-Euler-solutions/blob/master/haskell/p004.hs (written by Nayuki)
Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p004.java (written by Nayuki)
Mathematica: github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p004.mathematica (written by Nayuki)
C: github.com/eagletmt/project-euler-c/blob/master/1-9/problem4.c (written by eagletmt)
Go: github.com/frrad/project-euler/blob/master/golang/Problem004.go (written by Frederick Robinson)
Javascript: github.com/dsernst/ProjectEuler/blob/master/4 Largest palindrome product.js (written by David Ernst)
Scala: github.com/samskivert/euler-scala/blob/master/Euler004.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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51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75
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The 133 solved problems had an average difficulty of 16.9% at Project Euler and I scored 11,174 points (out of 12300) at Hackerrank's Project Euler+.
more about me can be found on my homepage.
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