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

# Problem 56: Powerful digit sum

(see projecteuler.net/problem=56)

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:

This is equivalent to`echo 10 | ./56`

Output:

*(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

see https://www.hackerrank.com/contests/projecteuler/challenges/euler056

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