<< problem 52 - Permuted multiples | Poker hands - problem 54 >> |
Problem 53: Combinatoric selections
(see projecteuler.net/problem=53)
There are exactly ten ways of selecting three from five, 12345:
123, 124, 125, 134, 135, 145, 234, 235, 245, and 345
In combinatorics, we use the notation, ^5C_3 = 10.
In general,
^nC_r = dfrac{n!}{r! * (n-r)!} where r <= n, n! = n * (n-1) * ... * 3 * 2 * 1, and 0! = 1.
It is not until n = 23, that a value exceeds one-million: ^{23}C_10 = 1144066.
How many, not necessarily distinct, values of ^nC_r, for 1 <= n <= 100, are greater than one-million?
My Algorithm
The formulas based on factorials (those provided in the problem statement) allow a direct computation of ^nC_r.
However, you might get very big numbers in the numerator and/or denominator - they easily exceed the range of a 32 or 64 bit integer.
There is another formula - the recursive definition:
- ^nC_0 = {^nC_n} = 1 and
- ^nC_k = {^{n-1}C_{k-1}} + {^{n-1}C_k}
maxNumber
, I set it to maxNumber + 1
. This value still fits into an 64 bit integer (even 2*(maxNumber+1)
is no problem).At the same time,
bigNumbers
is incremented.I am allowed to replace ^nC_k by
maxNumber + 1
because the true value of ^nC_k doesn't really matter - all we want to know is whether ^nC_k > maxNumber - or not.When all values are processed,
bigNumbers
is displayed.
Note
The program could use less memory: instead of storing all values of combinations[n][k]
it is sufficient to keep only
combinations[n-1][]
and combinations[n][]
, thus reducing the memory requirements from maxN^2 to 2*maxN.
Interactive test
You can submit your own input to my program and it will be instantly processed at my server:
This is equivalent toecho "23 1000000" | ./53
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.
#include <vector>
#include <iostream>
int main()
{
// maximum index n and/or k
unsigned int maxN = 100;
// what is considered "big" ?
unsigned long long maxNumber = 1000000;
std::cin >> maxN >> maxNumber;
// this will be the displayed result
unsigned int bigNumbers = 0;
// setup a 2D array to hold all values
std::vector<std::vector<unsigned long long>> combinations(maxN + 1);
// C(n,0) = C(n,n) = 1
for (unsigned int n = 0; n <= maxN; n++)
{
combinations[n].resize(n + 1, 0);
combinations[n][0] = combinations[n][n] = 1;
}
// recursive definition:
// C(n,k) = C(n-1, k-1) + C(n-1, k)
for (unsigned int n = 1; n <= maxN; n++)
for (unsigned int k = 1; k < n; k++)
{
auto sum = combinations[n - 1][k - 1] + combinations[n - 1][k];
// clamp numbers to avoid exceeding 64 bits
if (sum > maxNumber)
{
sum = maxNumber + 1;
// we found one more big number
bigNumbers++;
}
// store result
combinations[n][k] = sum;
}
std::cout << bigNumbers << std::endl;
return 0;
}
This solution contains 7 empty lines, 10 comments and 2 preprocessor commands.
Benchmark
The correct solution to the original Project Euler problem was found in less than 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 27, 2017 submitted solution
April 20, 2017 added comments
Hackerrank
see https://www.hackerrank.com/contests/projecteuler/challenges/euler053
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.
Links
projecteuler.net/thread=53 - the best forum on the subject (note: you have to submit the correct solution first)
Code in various languages:
C# www.mathblog.dk/project-euler-53-cnr-exceed-one-million/ (written by Kristian Edlund)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p053.java (written by Nayuki)
Javascript github.com/dsernst/ProjectEuler/blob/master/53 Combinatoric selections.js (written by David Ernst)
Go github.com/frrad/project-euler/blob/master/golang/Problem053.go (written by Frederick Robinson)
Mathematica github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p053.mathematica (written by Nayuki)
Haskell github.com/nayuki/Project-Euler-solutions/blob/master/haskell/p053.hs (written by Nayuki)
Scala github.com/samskivert/euler-scala/blob/master/Euler053.scala (written by Michael Bayne)
Those links are just an unordered selection of source code I found with a semi-automatic search script on Google/Bing/GitHub/whatever.
You will probably stumble upon better solutions when searching on your own. Maybe not all linked resources produce the correct result and/or exceed time/memory limits.
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 | |
the flashing problem is the one I solved most recently |
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I scored 13,486 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.
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 52 - Permuted multiples | Poker hands - problem 54 >> |