<< problem 124 - Ordered radicals | Cuboid layers - problem 126 >> |
Problem 125: Palindromic sums
(see projecteuler.net/problem=125)
The palindromic number 595 is interesting because it can be written as the sum of consecutive squares: 6^2 + 7^2 + 8^2 + 9^2 + 10^2 + 11^2 + 12^2.
There are exactly eleven palindromes below one-thousand that can be written as consecutive square sums, and the sum of these palindromes is 4164.
Note that 1 = 0^2 + 1^2 has not been included as this problem is concerned with the squares of positive integers.
Find the sum of all the numbers less than 108 that are both palindromic and can be written as the sum of consecutive squares.
My Algorithm
My function isPalindrome
determines whether it's parameter is a palindrome or not:
- iteratively remove the lowest digit and append it to
reverse
- if at the end
reverse == x
then we have a palindrome
Two nested loops iterate over all possible sequences.
My first solution wasn't correct because I didn't notice that sums may appear multipe times (there are two "collisions" for the original problem).
All palindromes are temporarily stored in
solutions
and then I remove all duplicates (see oeis.org/A267600).
Alternative Approaches
There is a closed formula for a sequence sum_{a=i..j}{a^2} but I'm not sure whether it would be faster because my incremental computation of current
is extremely efficient.
Modifications by HackerRank
The upper limit can be defined as well as the distance between consecutive "numbers" (what I call stride
):
instead of a^2 + (a+1)^2 + (a+2)^2 + ... the terms are a^2 + (a+stride)^2 + (a+2 * stride)^2 + ...
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 1000 1" | ./125
Output:
Note: the original problem's input 100000000 1
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 <vector>
#include <iostream>
#include <algorithm>
// return true if x is a palindrome
bool isPalindrome(unsigned int x)
{
auto reduced = x / 10;
auto reverse = x % 10;
// fast exit: a trailing zero can't create a palindrome
if (reverse == 0)
return false;
while (reduced > 0)
{
// chop off the lowest digit and append it to "reverse"
reverse *= 10;
reverse += reduced % 10;
reduced /= 10;
}
// palindrome ? both must be equal
return reverse == x;
}
int main()
{
unsigned int tests = 1;
std::cin >> tests;
while (tests--)
{
unsigned int limit = 100000000;
unsigned int stride = 1; // distance between consecutive square numbers
std::cin >> limit >> stride;
std::vector<unsigned int> solutions;
for (unsigned long long first = 1; 2*first*first < limit; first++)
{
auto next = first + stride;
// sum of a^2 + b^2 + ...
auto current = first * first + next * next;
// still within the limit ?
while (current < limit)
{
// check
if (isPalindrome(current))
solutions.push_back(current);
// add one element to the sequence
next += stride;
current += next * next;
}
}
// sort ...
std::sort(solutions.begin(), solutions.end());
// .. and remove duplicates
auto garbage = std::unique(solutions.begin(), solutions.end());
solutions.erase(garbage, solutions.end());
// count all solutions
unsigned long long sum = 0;
for (auto x : solutions)
sum += x;
std::cout << sum << std::endl;
}
return 0;
}
This solution contains 12 empty lines, 11 comments and 3 preprocessor commands.
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
May 17, 2017 submitted solution
May 17, 2017 added comments
Hackerrank
see https://www.hackerrank.com/contests/projecteuler/challenges/euler125
My code solves 35 out of 35 test cases (score: 100%)
Difficulty
Project Euler ranks this problem at 25% (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 rarely an option.
Links
projecteuler.net/thread=125 - 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-125-square-sums-palindromic/ (written by Kristian Edlund)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p125.java (written by Nayuki)
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 13,486 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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