<< problem 15 - Lattice paths | Number letter counts - problem 17 >> |

# Problem 16: Power digit sum

(see projecteuler.net/problem=16)

2^{15} = 32768 and the sum of its digits is 3 + 2 + 7 + 6 + 8 = 26.

What is the sum of the digits of the number 2^{1000}?

# My Algorithm

These two equations describe the iterative computation of 2^x:

2^0 = 1 and 2^x = 2 * 2^{x-1}

These numbers grow pretty big and for x=64 exceed the range of `unsigned long long`

.

That's why I store all decimal `Digits`

in a `std::vector`

, where the lowest index contains the least significant digits.

For x=0 - which represents 2^0 = 1 - this `Digits`

container's elements are `{ 1 }`

For x=15 - which represents 2^{15} = 1 - this `Digits`

container's elements are `{ 8, 6, 7, 2, 3 }`

Multiplying `Digits`

by 2 follows the same rules as basic multiplication taught in school:

1. multiply each digit by 2, start at the lowest digit

2. if digit * 2 >= 10 then an overflow occurred: carry over (digit * 2) div 10 to the next digit and keep (digit * 2) mod 10.

*Note:* we must carry over at most 1 and instead of a computationally expensive modulo we can subtract 10 which is faster

## Alternative Approaches

Exponentation by squaring saves many steps:

The result of 2^1024 can be found in just 10 steps instead of 1024.

## Modifications by HackerRank

To avoid timeouts, I store all results (even intermediate steps) in a cache and re-use as much as possible from this cache.

# Interactive test

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

This is equivalent to`echo "1 15" | ./16`

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.

#include <vector>
#include <iostream>
// store single digits in an array, lowest digit come first

typedef std::vector<unsigned int> Digits;
int main()
{
// memoize powers of two
std::vector<Digits> cache;
// add 2^0 = 1
cache.push_back({ 1 });
unsigned int tests;
std::cin >> tests;
while (tests--)
{
unsigned int exponent;
std::cin >> exponent;
// and compute the remaining exponents
for (unsigned int current = cache.size(); current <= exponent; current++)
{
auto power = cache.back();
unsigned int carry = 0;
for (auto& i : power)
{
// times two ...
i = 2 * i + carry;
// handle overflow
if (i >= 10)
{
i -= 10;
carry = 1;
}
else
{
carry = 0;
}
}
// still some carry left ?
if (carry != 0)
power.push_back(carry);
// memoize result
cache.push_back(power);
}
// sum of all digits
unsigned int sum = 0;
for (auto i : cache[exponent])
sum += i;
std::cout << sum << std::endl;
}
return 0;
}

This solution contains 9 empty lines, 9 comments and 2 preprocessor commands.

# Benchmark

The correct solution to the original Project Euler problem was found in less than 0.01 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 23, 2017 submitted solution

March 31, 2017 added comments

# Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler016

My code solves **10** out of **10** 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 20: Factorial 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=16 - **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-16/ (written by Kristian Edlund)

Haskell: github.com/nayuki/Project-Euler-solutions/blob/master/haskell/p016.hs (written by Nayuki)

Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p016.java (written by Nayuki)

Mathematica: github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p016.mathematica (written by Nayuki)

C: github.com/eagletmt/project-euler-c/blob/master/10-19/problem16.c (written by eagletmt)

Go: github.com/frrad/project-euler/blob/master/golang/Problem016.go (written by Frederick Robinson)

Javascript: github.com/dsernst/ProjectEuler/blob/master/16 Power digit sum.js (written by David Ernst)

Scala: github.com/samskivert/euler-scala/blob/master/Euler016.scala (written by Michael Bayne)

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.

# 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 |

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I scored 13,386 points (out of 15600 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 15 - Lattice paths | Number letter counts - problem 17 >> |