<< problem 201 - Subsets with a unique sum | Generalised Hamming Numbers - problem 204 >> |

# Problem 203: Squarefree Binomial Coefficients

(see projecteuler.net/problem=203)

The binomial coefficients ^nC_k can be arranged in triangular form, Pascal's triangle, like this:

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

.........

It can be seen that the first eight rows of Pascal's triangle contain twelve distinct numbers: 1, 2, 3, 4, 5, 6, 7, 10, 15, 20, 21 and 35.

A positive integer n is called squarefree if no square of a prime divides n. Of the twelve distinct numbers in the first eight rows of Pascal's triangle,

all except 4 and 20 are squarefree. The sum of the distinct squarefree numbers in the first eight rows is 105.

Find the sum of the distinct squarefree numbers in the first 51 rows of Pascal's triangle.

# My Algorithm

A number in Pascal's triangle in row n and column k has the value \binom{n}{k}.

There are two ways to compute this value:

1. \binom{n}{k} = frac{n!}{k!(n-k)!}

2. \binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}

The first equation tells me that each potential prime factor doesn't exceed 51 because the largest number contained in any of the factorials is 51.

Therefore `isSquarefree`

performs a trial division by p^2 where p are the prime numbers between 2 and 51.

(Actually I was too lazy to include a proper prime sieve and divide by all numbers between 2 and 51. There is no measureable performane loss.)

The second equation is useful to compute a row based on the previous row. This way I avoid potential overflows of large factorials, too.

The triangle is symmetric, therefore I check only the left half. And `unique`

keeps track of all numbers I have seen so far.

## Note

My code is far from optimal:

- `isSquarefree`

checks numbers which are not prime, too, which is redundant

- if a number appears multiple times in the triangles, `isSquarefree`

doesn't need to be called

# My code

… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too.

#include <iostream>
#include <vector>
#include <set>
// return true if x cannot be divided by p^2 for all primes 2 <= p <= maxSquare

bool isSquarefree(unsigned long long x, unsigned int maxSquare)
{
// instead of a proper prime sieve, just perform trial division of all numbers
for (unsigned int p = 2; p <= maxSquare; p++)
if (x % (p*p) == 0)
return false;
// yes, squarefree
return true;
}
int main()
{
// size of the triangle
unsigned int numRows = 51;
std::cin >> numRows;
// all squarefree numbers
std::set<unsigned long long> squareFree = { 1 };
// initial row
std::vector<unsigned long long> current = { 1 };
// ... and compute all further rows
for (unsigned int row = 1; row < numRows; row++)
{
// last and first element is always 1
std::vector<unsigned long long> next(current.size() + 1, 1);
// fill remaining cells
for (unsigned int column = 1; column < next.size() - 1; column++)
next[column] = current[column - 1] + current[column];
// symmetric: check only half of the triangle, skip borders, too (always 1)
for (unsigned int i = 1; i <= next.size() / 2; i++)
{
auto x = next[i];
if (isSquarefree(x, numRows))
squareFree.insert(x); // std::set prevents duplicates
}
// next iteration
current = std::move(next);
}
// find sum of all squarefree numbers
unsigned long long sum = 0;
for (auto x : squareFree)
sum += x;
// display result
std::cout << sum << std::endl;
return 0;
}

This solution contains 10 empty lines, 13 comments and 3 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 8 | ./203`

Output:

*Note:* the original problem's input `51`

__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 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

May 31, 2017 submitted solution

May 31, 2017 added comments

# Difficulty

Project Euler ranks this problem at **25%** (out of 100%).

# Links

projecteuler.net/thread=203 - **the** best forum on the subject (*note:* you have to submit the correct solution first)

Code in various languages:

Java: github.com/nayuki/Project-Euler-solutions/blob/master/java/p203.java (written by Nayuki)

# Heatmap

green problems solve the original Project Euler problem and have a perfect score of 100% at Hackerrank, too.

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

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

My username at Project Euler is

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# 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 201 - Subsets with a unique sum | Generalised Hamming Numbers - problem 204 >> |