<< 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?

# My 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).

# 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:

(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 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++)
{
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.

# Benchmark

The correct solution to the original Project Euler problem was found in 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
May 8, 2017 simplified code, it runs much faster now

# Hackerrank

My code solves 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 rarely 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.

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