<< problem 196 - Prime triplets | Iterative Circle Packing - problem 199 >> |
Problem 197: Investigating the behaviour of a recursively defined sequence
(see projecteuler.net/problem=197)
Given is the function f(x) = \lfloor 2^{30.403243784-x^2} \rfloor * 10^-9 ( \lfloor \space \rfloor is the floor-function),
the sequence u_n is defined by u_0 = -1 and u_{n+1} = f(u_n).
Find u_n + u_{n+1} for n = 10^12.
Give your answer with 9 digits after the decimal point.
My Algorithm
The function f(x)
is a straightforward implementation of the formula specified in the problem statement.
I printed the first results "to get a feeling" for the value. It's obvious that they quickly converge to a bi-stabil state.
Assuming that this doesn't change, I computed f(u_1000) + f(u_1001) and hoped for the best ... and yes, got the correct answer.
Actually you always get the correct answer after about 500 steps: u_512 + u_513 are the smallest on my machine.
Depending on some rounding issues it's possible that you need more or less steps on your computer.
Interactive test
This feature is not available for the current problem.
My code
… was written in C++ and can be compiled with G++, Clang++, Visual C++. You can download it, too.
#include <iostream>
#include <iomanip>
#include <cmath>
// compute next step
double f(double x)
{
return floor(pow(2.0, 30.403243784 - x*x)) * pow(10, -9);
}
int main()
{
// initial value
double u = -1;
// compute next steps
double next = f(u);
for (unsigned int i = 1; i < 513; i++)
{
u = next;
next = f(u);
}
// sum of two neighbors
double result = u + next;
std::cout << std::fixed << std::setprecision(9) << result << std::endl;
return 0;
}
This solution contains 5 empty lines, 4 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
June 13, 2017 submitted solution
June 13, 2017 added comments
Difficulty
Project Euler ranks this problem at 45% (out of 100%).
Links
projecteuler.net/thread=197 - the best forum on the subject (note: you have to submit the correct solution first)
Code in various languages:
Python github.com/HaochenLiu/My-Project-Euler/blob/master/197.py (written by Haochen Liu)
Python github.com/Meng-Gen/ProjectEuler/blob/master/197.py (written by Meng-Gen Tsai)
Python github.com/nayuki/Project-Euler-solutions/blob/master/python/p197.py (written by Nayuki)
Python github.com/smacke/project-euler/blob/master/python/197.py (written by Stephen Macke)
C++ github.com/roosephu/project-euler/blob/master/197.cpp (written by Yuping Luo)
C github.com/LaurentMazare/ProjectEuler/blob/master/e197.c (written by Laurent Mazare)
Java github.com/dcrousso/ProjectEuler/blob/master/PE197.java (written by Devin Rousso)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p197.java (written by Nayuki)
Java github.com/thrap/project-euler/blob/master/src/Java/Problem197.java (written by Magnus Solheim Thrap)
Go github.com/frrad/project-euler/blob/master/golang/Problem197.go (written by Frederick Robinson)
Mathematica github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p197.mathematica (written by Nayuki)
Mathematica github.com/steve98654/ProjectEuler/blob/master/197.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 196 - Prime triplets | Iterative Circle Packing - problem 199 >> |