<< problem 3 - Largest prime factor Smallest multiple - problem 5 >>

# Problem 4: Largest palindrome product

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:

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

# Hackerrank

My code solves 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 rarely an option.

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)
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 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.

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

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The 206 solved problems had an average difficulty of 27.5% at Project Euler and
I scored 12,626 points (out of 14300 possible points, top rank was 20 out ouf ≈60000 in July 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.

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