<< problem 135 - Same differences | Fibonacci golden nuggets - problem 137 >> |
Problem 136: Singleton differences
(see projecteuler.net/problem=136)
The positive integers, x, y, and z, are consecutive terms of an arithmetic progression.
Given that n is a positive integer, the equation, x^2 - y^2 - z^2 = n, has exactly one solution when n = 20:
13^2 - 10^2 - 7^2 = 20
In fact there are twenty-five values of n below one hundred for which the equation has a unique solution.
How many values of n less than fifty million have exactly one solution?
My Algorithm
My solution is almost identical to problem 135. See there for an explanation of the algorithm.
I actually submitted the correct solution to this problem just four minutes after.
Then I thought: let's reduce the memory consumption ! The code from problem 135 required about 200 MByte RAM.
I replaced the std::vector<unsigned int> solutions
by two bitfields atLeastOne
and moreThanOne
:
- both have one bit per possible solution
- both are initialized with false
- if
atLeastOne[current]
isfalse
, then there was no solution so far → set it totrue
- if
atLeastOne[current]
istrue
, then there was at least one solution → setmoreThanOne[current]
totrue
true
entries in atLeastOne
minus all true
in moreThanOne
.Memory consumption is reduced to 1/16th ... or 14 MByte including I/O overhead etc. Execution time remained unchanged.
Interactive test
This feature is not available for the current problem.
My code
… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too. Or just jump to my GitHub repository.
#include <iostream>
#include <vector>
int main()
{
const unsigned int limit = 50000000;
// all bits set to zero
std::vector<bool> atLeastOne (limit, false);
std::vector<bool> moreThanOne(limit, false);
for (unsigned int a = 1; a < limit; a++)
for (auto b = (a + 3) / 4; b < a; b++)
{
auto current = a * (4*b - a);
if (current >= limit)
break;
// already had a solution (or more) ?
if (atLeastOne[current])
moreThanOne[current] = true;
else
// nope, first solution
atLeastOne [current] = true;
}
// count all with exactly 1 solution
unsigned int count = 0;
for (unsigned int i = 0; i < limit; i++)
if (atLeastOne[i] && !moreThanOne[i])
count++;
std::cout << count << std::endl;
return 0;
}
This solution contains 6 empty lines, 4 comments and 2 preprocessor commands.
Benchmark
The correct solution to the original Project Euler problem was found in 1.2 seconds on an Intel® Core™ i7-2600K CPU @ 3.40GHz.
Peak memory usage was about 14 MByte.
(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 19, 2017 submitted solution
May 22, 2017 added comments
Difficulty
Project Euler ranks this problem at 45% (out of 100%).
Links
projecteuler.net/thread=136 - 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-136-singleton-difference/ (written by Kristian Edlund)
C# github.com/HaochenLiu/My-Project-Euler/blob/master/136.cs (written by Haochen Liu)
C++ github.com/Meng-Gen/ProjectEuler/blob/master/136.cc (written by Meng-Gen Tsai)
C++ github.com/smacke/project-euler/blob/master/cpp/136.cpp (written by Stephen Macke)
C++ github.com/steve98654/ProjectEuler/blob/master/136.cpp
C github.com/zmwangx/Project-Euler/blob/master/136/136.c (written by Zhiming Wang)
Java github.com/dcrousso/ProjectEuler/blob/master/PE136.java (written by Devin Rousso)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p136.java (written by Nayuki)
Java github.com/thrap/project-euler/blob/master/src/Java/Problem136.java (written by Magnus Solheim Thrap)
Go github.com/frrad/project-euler/blob/master/golang/Problem136.go (written by Frederick Robinson)
Mathematica github.com/steve98654/ProjectEuler/blob/master/136.nb
Perl github.com/shlomif/project-euler/blob/master/project-euler/136/euler-136.pl (written by Shlomi Fish)
Rust github.com/gifnksm/ProjectEulerRust/blob/master/src/bin/p136.rs
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 | |
[new] | the flashing problem is the one I solved most recently |
I stopped working on Project Euler problems around the time they released 617.
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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 !!!
<< problem 135 - Same differences | Fibonacci golden nuggets - problem 137 >> |