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# Problem 25: 1000-digit Fibonacci number

The Fibonacci sequence is defined by the recurrence relation:

F_n = F_{n-1} + F_{n-2}, where F_1 = 1 and F_2 = 1.

Hence the first 12 terms will be:

F_1 = 1
F_2 = 1
F_3 = 2
F_4 = 3
F_5 = 5
F_6 = 8
F_7 = 13
F_8 = 21
F_9 = 34
F_{10} = 55
F_{11} = 89
F_{12} = 144

The 12th term, F_{12}, is the first term to contain three digits.

What is the index of the first term in the Fibonacci sequence to contain 1000 digits?

# Algorithm

I precompute all Fibonacci number with up to 5000 digits (a design decision influenced by Hackerrank's modified problem) and keep those results in cache.

Unfortunately, there is a small problem with C++ ...
F_{47}=2971215073 is the largest Fibonacci number that fits in a 32-bit integer and
F_{94}=19740274219868223167 is too big for a 64-bit integer.

My program stores such large number as a std::vector where index 0 contains the least significant digit ("in reverse order").
E.g. F_{23}=28657 is represented as { 7, 5, 6, 8, 2 }

The function add returns the sum of two large numbers a and b where b>=a.
The algorithm behind this function is exactly the same you were taught in primary school.

## Alternative Approaches

The main problem was adding two very large numbers. When programming in Python, Java, etc. you get these things for free.

## Modifications by HackerRank

The large amount of test cases was the main cause for dividing my solution into two parts;
1. precompute all relevant Fibonacci numbers (done once - "expensive")
2. look up the result (performed many, many times - "cheap")

# 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>

// store single digits because numbers get too big for C++
typedef std::vector<unsigned int> Digits;

// Hackerrank's upper limit
const unsigned int MaxDigits = 5000;

// add two long number where b >= a
Digits add(const Digits& a, const Digits& b)
{
Digits result = b;

unsigned int carry = 0;
for (unsigned int i = 0; i < result.size(); i++)
{
// "a" might have less digits than "b"
if (i < a.size())
result[i] += a[i];

// don't forget about the carry ...
result[i] += carry;

// handle overflow
if (result[i] >= 10)
{
carry = 1;
result[i] -= 10;
}
else
carry = 0;
}

// largest digit not overflowing ?
if (carry != 0)
result.push_back(carry);

return result;
}

int main()
{
// precompute number of steps we needed for each number of digits
// [number of digits] => [index of smallest Fibonacci number]
std::vector<unsigned int> cache = { 0, 1 }; // F_0 is undefined
cache.reserve(MaxDigits);

// f(1) = 1
Digits a = { 1 };
// f(2) = 1
Digits b = { 1 };
// we have predefined F_1 and F_2
unsigned int fiboIndex = 2;

while (cache.size() <= MaxDigits)
{
// next Fibonacci number
fiboIndex++;
a = std::move(b);
b = std::move(next);

// digits of current Fibonacci number
auto numDigits = b.size();
// digits of the previously largest Fibonacci number
auto largestKnown = cache.size() - 1; // don't count the 0th element

// one more digit than before ?
if (largestKnown < numDigits)
cache.push_back(fiboIndex);
}

// simply look up the result
unsigned int tests;
std::cin >> tests;
while (tests--)
{
unsigned int numDigits;
std::cin >> numDigits;
std::cout << cache[numDigits] << std::endl;
}

return 0;
}


This solution contains 15 empty lines, 17 comments and 2 preprocessor commands.

# Interactive test

You can submit your own input to my program and it will be instantly processed at my server:

Number of test cases (1-5):

Input data (separated by spaces or newlines):

This is equivalent to
echo "1 3" | ./25

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)

# Benchmark

The correct solution to the original Project Euler problem was found in 0.41 seconds on a Intel® Core™ i7-2600K CPU @ 3.40GHz.
(compiled for x86_64 / Linux, GCC flags: -O3 -march=native -fno-exceptions -fno-rtti -std=c++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 24, 2017 submitted solution

# Hackerrank

My code solved 4 out of 4 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 never an option.

projecteuler.net/thread=25 - the best forum on the subject (note: you have to submit the correct solution first)

Code in various languages:

Python: www.mathblog.dk/project-euler-25-fibonacci-sequence-1000-digits/ (written by Kristian Edlund)
Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p025.java (written by Nayuki)
Mathematica: github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p025.mathematica (written by Nayuki)
C: github.com/eagletmt/project-euler-c/blob/master/20-29/problem25.c (written by eagletmt)
Javascript: github.com/dsernst/ProjectEuler/blob/master/25 1000-digit Fibonacci number.js (written by David Ernst)
Scala: github.com/samskivert/euler-scala/blob/master/Euler025.scala (written by Michael Bayne)

# Heatmap

green problems solve the original Project Euler problem and have a perfect score of 100% at Hackerrank, too.
yellow problems score less than 100% at Hackerrank (but still solve the original problem).
gray problems are already solved but I haven't published my solution yet.
blue problems are already solved and there wasn't a Hackerrank version of it (at the time I solved it) or I didn't care about it because it differed too much.

Please click on a problem's number to open my solution to that problem:

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The 133 solved problems had an average difficulty of 16.9% at Project Euler and I scored 11,174 points (out of 12300) at Hackerrank's Project Euler+.
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