<< problem 56 - Powerful digit sum | Spiral primes - problem 58 >> |
Problem 57: Square root convergents
(see projecteuler.net/problem=57)
It is possible to show that the square root of two can be expressed as an infinite continued fraction.
sqrt{2} = 1 + frac{1}{2 + frac{1}{2 + frac{1}{2 + ... }}} = 1.414213...
By expanding this for the first four iterations, we get:
1 + dfrac{1}{2} = dfrac{3}{2} = 1.5
1 + dfrac{1}{2 + frac{1}{2}} = dfrac{7}{5} = 1.4
1 + dfrac{1}{2 + frac{1}{2 + frac{1}{2}}} = dfrac{17}{12} = 1.41666...
1 + dfrac{1}{2 + frac{1}{2 + frac{1}{2 + frac{1}{2}}}} = dfrac{41}{29} = 1.41379...
The next three expansions are dfrac{99}{70} , dfrac{239}{169}, and dfrac{577}{408}, but the eighth expansion, dfrac{1393}{985},
is the first example where the number of digits in the numerator exceeds the number of digits in the denominator.
In the first one-thousand expansions, how many fractions contain a numerator with more digits than denominator?
My Algorithm
An iteration/expansion can be described as:
f(n+1) = 1 + dfrac{1}{2 + f(n)}
Each iteration/expansion is a fraction, let's call it f(n) = dfrac{a(n)}{b(n)}
f(n+1) = 1 + dfrac{1}{1 + frac{a(n)}{b(n)}}
= 1 + dfrac{1}{frac{b(n) + a(n)}{b(n)}}
= 1 + dfrac{b(n)}{b(n) + a(n)}
= dfrac{b(n) + a(n) + b(n)}{b(n) + a(n)}
dfrac{a(n+1)}{b(n+1)} = dfrac{2b(n) + a(n)}{b(n) + a(n)}
so it's actually pretty easy to continuously compute numerator and denominator:
a(n+1) = 2b(n) + a(n)
b(n+1) = b(n) + a(n)
inital values:
x(0) = 1 + 1/2 = 1 + 1/(1+1)
a(0) = 1
b(0) = 1
The BigNum
was copied from problem 56. When MaxDigit = 10
then each element of the array is one digit and I can compare a.size() > b.size()
.
Modifications by HackerRank
I have to print each iteration's ID where the numerator has more digits than the denominator.
Interactive test
You can submit your own input to my program and it will be instantly processed at my server:
This is equivalent toecho 14 | ./57
Output:
Note: the original problem's input 1000
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. Or just jump to my GitHub repository.
The code contains #ifdef
s to switch between the original problem and the Hackerrank version.
Enable #ifdef ORIGINAL
to produce the result for the original problem (default setting for most problems).
#include <vector>
#include <iostream>
// store single digits with lowest digits first
// e.g. 1024 is stored as { 4,2,0,1 }
// only non-negative numbers supported
struct BigNum : public std::vector<unsigned int>
{
// must be 10 for this problem: a single "cell" store one digit 0 <= digit < 10
static const unsigned int MaxDigit = 10;
// store a non-negative number
BigNum(unsigned long long x = 0)
{
do
{
push_back(x % MaxDigit);
x /= MaxDigit;
} while (x > 0);
}
// add two big numbers
BigNum operator+(const BigNum& other) const
{
auto result = *this;
// add in-place, make sure it's big enough
if (result.size() < other.size())
result.resize(other.size(), 0);
unsigned int carry = 0;
for (size_t i = 0; i < result.size(); i++)
{
carry += result[i];
if (i < other.size())
carry += other[i];
else
if (carry == 0)
return result;
if (carry < MaxDigit)
{
// no overflow
result[i] = carry;
carry = 0;
}
else
{
// yes, we have an overflow
result[i] = carry - MaxDigit;
carry = 1;
}
}
if (carry > 0)
result.push_back(carry);
return result;
}
};
#define ORIGINAL
int main()
{
unsigned int iterations = 1000;
std::cin >> iterations;
// both values have one digit initialized with 1
BigNum a = 1;
BigNum b = 1;
unsigned int count = 0;
for (unsigned int i = 0; i <= iterations; i++)
{
// different number of digits ?
if (a.size() > b.size())
{
#ifdef ORIGINAL
count++;
#else
std::cout << i << std::endl;
#endif
}
// a(n+1) = 2*b(n) + a(n)
// b(n+1) = b(n) + a(n)
auto twoB = b + b;
auto nextA = a + twoB;
auto nextB = b + a;
a = std::move(nextA);
b = std::move(nextB);
}
#ifdef ORIGINAL
std::cout << count << std::endl;
#endif
return 0;
}
This solution contains 15 empty lines, 13 comments and 8 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
February 28, 2017 submitted solution
April 24, 2017 added comments
Hackerrank
see https://www.hackerrank.com/contests/projecteuler/challenges/euler057
My code solves 8 out of 8 test cases (score: 100%)
Difficulty
Project Euler ranks this problem at 5% (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.
Similar problems at Project Euler
Problem 56: Powerful digit sum
Note: I'm not even close to solving all problems at Project Euler. Chances are that similar problems do exist and I just haven't looked at them.
Links
projecteuler.net/thread=57 - 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-57-square-root-two/ (written by Kristian Edlund)
Python github.com/hughdbrown/Project-Euler/blob/master/euler-057.py (written by Hugh Brown)
Python github.com/Meng-Gen/ProjectEuler/blob/master/57.py (written by Meng-Gen Tsai)
Python github.com/nayuki/Project-Euler-solutions/blob/master/python/p057.py (written by Nayuki)
C++ github.com/HaochenLiu/My-Project-Euler/blob/master/057.cpp (written by Haochen Liu)
Java github.com/dcrousso/ProjectEuler/blob/master/PE057.java (written by Devin Rousso)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p057.java (written by Nayuki)
Java github.com/thrap/project-euler/blob/master/src/Java/Problem57.java (written by Magnus Solheim Thrap)
Go github.com/frrad/project-euler/blob/master/golang/Problem057.go (written by Frederick Robinson)
Mathematica github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p057.mathematica (written by Nayuki)
Mathematica github.com/steve98654/ProjectEuler/blob/master/057.nb
Haskell github.com/roosephu/project-euler/blob/master/57.hs (written by Yuping Luo)
Clojure github.com/guillaume-nargeot/project-euler-clojure/blob/master/src/project_euler/problem_057.clj (written by Guillaume Nargeot)
Clojure github.com/rm-hull/project-euler/blob/master/src/euler057.clj (written by Richard Hull)
Scala github.com/samskivert/euler-scala/blob/master/Euler057.scala (written by Michael Bayne)
Perl github.com/gustafe/projecteuler/blob/master/057-Square-root-convergents.pl (written by Gustaf Erikson)
Rust github.com/gifnksm/ProjectEulerRust/blob/master/src/bin/p057.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 56 - Powerful digit sum | Spiral primes - problem 58 >> |