<< problem 55 - Lychrel numbers Square root convergents - problem 57 >>

# Problem 56: Powerful digit sum

A googol (10^100) is a massive number: one followed by one-hundred zeros; 100^100 is almost unimaginably large: one followed by two-hundred zeros.
Despite their size, the sum of the digits in each number is only 1.

Considering natural numbers of the form, ab, where a, b < 100, what is the maximum digital sum?

# Algorithm

I wrote a small class BigNum that handles arbitrarily large integers (only positive, no sign).
It supports multiplication based on the simple algorithm that you'd use with pen and paper, too.

My inner loop has a variable called power which represents base^{exponent}. Then base^{exponent+1} = base^{exponent} * base.
The digit sum iterates over all digits and keeps track of the largest sum (maxSum).

# 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 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)
{
// actually the constructor is always called with x = 1, but I keep my default implementation
do
{
push_back(x % MaxDigit);
x /= MaxDigit;
} while (x > 0);
}

// multiply a big number by an integer
BigNum operator*(unsigned int factor) const
{
unsigned long long carry = 0;
auto result = *this;
// multiply each block by the number, take care of temporary overflows (carry)
for (auto& i : result)
{
carry += i * (unsigned long long)factor;
i      = carry % MaxDigit;
carry /= MaxDigit;
}
// store remaining carry in new digits
while (carry > 0)
{
result.push_back(carry % MaxDigit);
carry /= MaxDigit;
}

return result;
}
};

int main()
{
// maximum base/exponent (100 for Googol)
unsigned int maximum = 100;
std::cin >> maximum;

// look at all i^j
unsigned int maxSum = 1;
for (unsigned int base = 1; base <= maximum; base++)
{
// incrementally compute base^exponent
BigNum power = 1;
for (unsigned int exponent = 1; exponent <= maximum; exponent++)
{
// add all digits
unsigned int sum = 0;
for (auto digit : power)
sum += digit;

// new world record ? ;-)
if (maxSum < sum)
maxSum = sum;

// same base, next exponent:
// base^(exponent + 1) = (base^exponent) * base
power = power * base;
}
}

std::cout << maxSum << std::endl;
return 0;
}


This solution contains 9 empty lines, 16 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:

Input data (separated by spaces or newlines):

This is equivalent to
echo 10 | ./56

Output:

(please click 'Go !')

(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 0.01 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 28, 2017 submitted solution
April 24, 2017 added comments
May 8, 2017 simplified code, it runs much faster now

# Hackerrank

My code solved 5 out of 5 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.

# Similar problems at Project Euler

Problem 57: Square root convergents

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=56 - 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-56-considering-natural-numbers-of-the-form-ab-finding-the-maximum-digital-sum/ (written by Kristian Edlund)
Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p056.java (written by Nayuki)
Scala: github.com/samskivert/euler-scala/blob/master/Euler056.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+.
 << problem 55 - Lychrel numbers Square root convergents - problem 57 >>
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