<< problem 87 - Prime power triples | Roman numerals - problem 89 >> |

# Problem 88: Product-sum numbers

(see projecteuler.net/problem=88)

A natural number, N, that can be written as the sum and product of a given set of at least two natural numbers, {a1, a2, ... , ak}

is called a product-sum number: N = a_1 + a_2 + ... + a_k = a_1 * a_2 * ... * a_k.

For example, 6 = 1 + 2 + 3 = 1 * 2 * 3.

For a given set of size, k, we shall call the smallest N with this property a minimal product-sum number.

The minimal product-sum numbers for sets of size, k = 2, 3, 4, 5, and 6 are as follows.

k=2: 4 = 2 * 2 = 2 + 2

k=3: 6 = 1 * 2 * 3 = 1 + 2 + 3

k=4: 8 = 1 * 1 * 2 * 4 = 1 + 1 + 2 + 4

k=5: 8 = 1 * 1 * 2 * 2 * 2 = 1 + 1 + 2 + 2 + 2

k=6: 12 = 1 * 1 * 1 * 1 * 2 * 6 = 1 + 1 + 1 + 1 + 2 + 6

Hence for 2 <= k <= 6, the sum of all the minimal product-sum numbers is 4+6+8+12 = 30; note that 8 is only counted once in the sum.

In fact, as the complete set of minimal product-sum numbers for 2 <= k <= 12 is {4, 6, 8, 12, 15, 16}, the sum is 61.

What is the sum of all the minimal product-sum numbers for 2 <= k <= 12000?

# My Algorithm

My program is based on a Dynamic Programming approach, the core routine is `getMinK`

:

- try all divisors `i`

of a number `n`

, remove them from the product and sum and call `getMinK`

recursively: `getMinK(n, product / i, sum - i)`

- initially `product = n`

and `sum = n`

I had to add a few optimizations:

- if the product is reduced to `1`

and the sum is still bigger than `1`

then remaining `sum`

must be a sequence of 1s because 1 * product = product

- if `product == sum`

then the last factor/summand was found, but we can't do that in the first iteration (because the product / sum need at least two terms)

- if I removed `i`

, then I already tried all potential divisors smaller than `i`

→ keep track of `i`

with the parameter `minFactor`

# My code

… was written in C++ and can be compiled with G++, Clang++, Visual C++. You can download it, too.

#include <iostream>
#include <vector>
// initial values of the generated sequence:
// oeis.org/A104173

// max. number of factors / summands

const unsigned int Limit = 200000; // Project Euler: 12000
// [k] => [size of set]

std::vector<unsigned int> minK(Limit, 9999999); // initialize with an extremely high value
// found a better solution with k terms for number n ?

bool valid(unsigned int n, unsigned int k)
{
// too many terms ? (more than 12000)
if (k >= minK.size())
return false;
// found a smaller number n with the same number of terms k ?
if (minK[k] > n)
{
// note: the default value of minK[] is 9999999
minK[k] = n;
return true;
}
return false;
}
// return number of minimum k found (a number may be responsible for multiple minimum k, e.g. 8 => k=4 and k=5)
// n: the initial number
// product: n divided by removed numbers
// sum: n minus removed numbers
// depth: count removed numbers
// minFactor: skip checking factors smalled than this

unsigned int getMinK(unsigned int n, unsigned int product, unsigned int sum,
unsigned int depth = 1, unsigned int minFactor = 2)
{
// already removed all factors > 1 ?
// => add a bunch of 1s to the sum
if (product == 1)
return valid(n, depth + sum - 1) ? 1 : 0;
unsigned int result = 0;
if (depth > 1)
{
// perfect match ?
if (product == sum)
return valid(n, depth) ? 1 : 0;
// try to finish sequence
if (valid(n, depth + sum - product))
result++;
}
// and now all remaining factors
for (unsigned int i = minFactor; i*i <= product; i++)
if (product % i == 0) // found a factor ? let's dig deeper ...
result += getMinK(n, product / i, sum - i, depth + 1, i);
return result;
}
int main()
{
unsigned int limit;
std::cin >> limit;
minK.resize(limit + 1);
// k(2) = 4
unsigned int n = 4;
// result
unsigned int sum = 0;
// 0 and 1 are not used, still 11999 to go ...
unsigned int todo = limit - 1;
while (todo > 0)
{
// analyze n
unsigned int found = getMinK(n, n, n);
// at least one new k(n) found ?
if (found > 0)
{
todo -= found;
sum += n;
}
// next number
n++;
}
// print result
std::cout << sum << std::endl;
return 0;
}

This solution contains 16 empty lines, 26 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 50 | ./88`

Output:

*Note:* the original problem's input `50000000`

__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)*

# 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

March 19, 2017 submitted solution

May 5, 2017 added comments

# Hackerrank

see https://www.hackerrank.com/contests/projecteuler/challenges/euler088

My code solves **41** out of **41** test cases (score: **100%**)

# Difficulty

Project Euler ranks this problem at **40%** (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.

# Links

projecteuler.net/thread=88 - **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-88-minimal-product-sum-numbers/ (written by Kristian Edlund)

Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p088.java (written by Nayuki)

Scala: github.com/samskivert/euler-scala/blob/master/Euler088.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 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.

red problems are solved but exceed the time limit of one minute or the memory limit of 256 MByte.

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I scored 13,183 points (out of 15300 possible points, top rank was 17 out of ≈60000 in August 2017) at Hackerrank's Project Euler+.

Look at my progress and performance pages to get more details.

My username at Project Euler is

**stephanbrumme**while it's stbrumme at Hackerrank.

# 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 87 - Prime power triples | Roman numerals - problem 89 >> |