<< problem 24 - Lexicographic permutations Reciprocal cycles - problem 26 >>

# 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?

# My 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")

# 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)

# 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 <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.

# Benchmark

The correct solution to the original Project Euler problem was found in 0.4 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 24, 2017 submitted solution

# Hackerrank

My code solves 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 rarely an option.

# 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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The 310 solved problems (that's level 12) had an average difficulty of 32.6% at Project Euler and
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.

 << problem 24 - Lexicographic permutations Reciprocal cycles - problem 26 >>
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