<< problem 3 - Largest prime factor | Smallest multiple - problem 5 >> |
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.
My 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.
Interactive test
You can submit your own input to my program and it will be instantly processed at my server:
This is equivalent toecho "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)
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.
Benchmark
The correct solution to the original Project Euler problem was found in less than 0.01 seconds on an Intel® Core™ i7-2600K CPU @ 3.40GHz.
(compiled for x86_64 / Linux, GCC flags: -O3 -march=native -fno-exceptions -fno-rtti -std=gnu++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 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.
Links
projecteuler.net/thread=4 - the best forum on the subject (note: you have to submit the correct solution first)
Code in various languages:
C# www.mathblog.dk/project-euler-problem-4/ (written by Kristian Edlund)
C github.com/eagletmt/project-euler-c/blob/master/1-9/problem4.c (written by eagletmt)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p004.java (written by Nayuki)
Javascript github.com/dsernst/ProjectEuler/blob/master/4 Largest palindrome product.js (written by David Ernst)
Go github.com/frrad/project-euler/blob/master/golang/Problem004.go (written by Frederick Robinson)
Mathematica github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p004.mathematica (written by Nayuki)
Haskell github.com/nayuki/Project-Euler-solutions/blob/master/haskell/p004.hs (written by Nayuki)
Scala github.com/samskivert/euler-scala/blob/master/Euler004.scala (written by Michael Bayne)
Those links are just an unordered selection of source code I found with a semi-automatic search script on Google/Bing/GitHub/whatever.
You will probably stumble upon better solutions when searching on your own. Maybe not all linked resources produce the correct result and/or exceed time/memory limits.
Heatmap
Please click on a problem's number to open my solution to that problem:
green | solutions solve the original Project Euler problem and have a perfect score of 100% at Hackerrank, too | |
yellow | solutions score less than 100% at Hackerrank (but still solve the original problem easily) | |
gray | problems are already solved but I haven't published my solution yet | |
blue | solutions are relevant for Project Euler only: there wasn't a Hackerrank version of it (at the time I solved it) or it differed too much | |
orange | problems are solved but exceed the time limit of one minute or the memory limit of 256 MByte | |
red | problems are not solved yet but I wrote a simulation to approximate the result or verified at least the given example - usually I sketched a few ideas, too | |
black | problems are solved but access to the solution is blocked for a few days until the next problem is published | |
the flashing problem is the one I solved most recently |
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I scored 13526 points (out of 15700 possible points, top rank was 17 out of ≈60000 in August 2017) at Hackerrank's Project Euler+.
My username at Project Euler is stephanbrumme while it's stbrumme at Hackerrank.
Look at my progress and performance pages to get more details.
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 !!!
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