<< problem 234 - Semidivisible numbers |
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Tours on a 4 x n playing board - problem 237 >> |
Problem 235: An Arithmetic Geometric sequence
(see projecteuler.net/problem=235)
Given is the arithmetic-geometric sequence u(k) = (900-3k)r^{k-1}.
Let s(n) = sum_{k=1...n}u(k).
Find the value of r for which s(5000) = -600,000,000,000.
Give your answer rounded to 12 places behind the decimal point.
My Algorithm
The function s_r(5000) is monotonically decreasing: larger r reduce s_r(5000).
My program was inspired by the a classic bisection algorithm (en.wikipedia.org/wiki/Bisection_method):
1. start with two reasonable upper and lower limits (I choose 0 and 2)
2. find mid = frac{upper+lower}{2} and s_{mid}(5000)
3. if the result is less than -600 billion then mid is too large and set upper = mid
4. if the result is greater than -600 billion then mid is too small and set lower = mid
5. if upper - lower > 10^-12 then go to step 2.
The pow
function can be pretty slow. Incrementally computing r^k is about ten times faster (see #define SLOW
).
Note
I actually stop only after upper - lower < 10^-13 because I was afraid of rounding issues.
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 <iomanip>
#include <cmath>
//#define SLOW
// compute s(r) according to problem statement
double s(double r)
{
double result = 0;
double x = 1; // r^0 is always 0
for (int k = 1; k <= 5000; k++)
{
#ifdef SLOW
result += (900 - 3 * k) * pow(r, k - 1);
#else
result += (900 - 3 * k) * x;
x *= r;
#endif
}
return result;
}
int main()
{
// initial lower and upper bounds
double lower = 0;
double upper = 2;
// until the range is small enough
while (upper - lower > 0.0000000000001)
{
double mid = (upper + lower) / 2;
// find result at midpoint
double current = s(mid);
// adjust borders
if (current < -600000000000.0)
upper = mid;
else
lower = mid;
}
// print result (use midpoint but upper and/or lower should yield the same output)
std::cout << std::fixed << std::setprecision(12) << (upper + lower) / 2 << std::endl;
return 0;
}
This solution contains 6 empty lines, 7 comments and 6 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
July 2, 2017 submitted solution
July 2, 2017 added comments
Difficulty
Project Euler ranks this problem at 40% (out of 100%).
Links
projecteuler.net/thread=235 - the best forum on the subject (note: you have to submit the correct solution first)
Code in various languages:
Python github.com/smacke/project-euler/blob/master/python/235.py (written by Stephen Macke)
C++ github.com/roosephu/project-euler/blob/master/235.cpp (written by Yuping Luo)
C github.com/LaurentMazare/ProjectEuler/blob/master/e235.c (written by Laurent Mazare)
Java github.com/thrap/project-euler/blob/master/src/Java/Problem235.java (written by Magnus Solheim Thrap)
Go github.com/frrad/project-euler/blob/master/golang/Problem235.go (written by Frederick Robinson)
Mathematica github.com/steve98654/ProjectEuler/blob/master/235.nb
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 234 - Semidivisible numbers |
![]() |
Tours on a 4 x n playing board - problem 237 >> |