<< problem 249 - Prime Subset Sums | Reachable Numbers - problem 259 >> |
Problem 250: 250250
(see projecteuler.net/problem=250)
Find the number of non-empty subsets of {11, 22, 33,..., 250250^250250}, the sum of whose elements is divisible by 250.
Enter the rightmost 16 digits as your answer.
My Algorithm
The efficient powmod
comes from my toolbox.
The main algorithm of my solution is actually pretty similar to the previous problem (see problem 249).
A major difference is that I need a second array last
because there is no way to change values in-place without
falsifying previous results (due to the pseudo-random pattern of the residue).
At the end each value of current[i]
contains the number of set with residue i
(mod 10^16).
Those divisible by 250 have residue 0 and are found in current[0]
.
Interactive test
You can submit your own input to my program and it will be instantly processed at my server:
This is equivalent toecho "100 100" | ./250
Output:
Note: the original problem's input 250250 250
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)
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>
// powmod's code was taken from my toolbox
unsigned int powmod(unsigned int base, unsigned int exponent, unsigned int modulo)
{
unsigned int result = 1;
while (exponent > 0)
{
// fast exponentation:
// odd exponent ? a^b = a*a^(b-1)
if (exponent & 1)
result = (result * (unsigned long long)base) % modulo;
// even exponent ? a^b = (a*a)^(b/2)
base = (base * (unsigned long long)base) % modulo;
exponent >>= 1;
}
return result;
}
int main()
{
// from 1 to 250250
unsigned int limit = 250250;
// numbers divisible by 250 => residue/remainder is zero
unsigned int modulo = 250;
std::cin >> limit >> modulo;
// compute all (i^i) % 250
std::vector<unsigned int> residues;
for (unsigned int i = 0; i <= limit; i++)
residues.push_back(powmod(i, i, modulo));
// 10^16
const auto SixteenDigits = 10000000000000000ULL;
// how many sets of the current size have which residue
std::vector<unsigned long long> current(modulo + 1, 0);
// the empty set is divisible by 250 (and there's only one empty set)
current[0] = 1;
// and the same for all sets with one more element
auto last = current;
// step-by-step add each number ...
for (unsigned int i = 1; i <= limit; i++)
{
// ... to each set
for (unsigned int j = 0; j < modulo; j++)
{
// only possible if there was at least one set
if (last[j] == 0) // these two lines can be removed, too,
continue; // they are no major performance improvement
// how does the new number change the residue of the set ?
auto newResidue = residues[i] + j;
// wrap around
if (newResidue >= modulo) // same as newResidue %= modulo but a bit faster
newResidue -= modulo;
// add the number of all "old" sets to the new residue
current[newResidue] += last[j];
// keep only the last 16 digits
current[newResidue] %= SixteenDigits;
}
// copy data for next iteration
last = current;
}
// final value is the first element (where residue is 0 => divisible by 250)
auto result = current[0];
// ... but don't count the empty set
if (result == 0)
result = SixteenDigits;
result--;
// and display it
std::cout << result << std::endl;
return 0;
}
This solution contains 14 empty lines, 22 comments and 2 preprocessor commands.
Benchmark
The correct solution to the original Project Euler problem was found in 0.27 seconds on an Intel® Core™ i7-2600K CPU @ 3.40GHz.
Peak memory usage was about 3 MByte.
(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
July 14, 2017 submitted solution
July 14, 2017 added comments
Difficulty
Project Euler ranks this problem at 55% (out of 100%).
Links
projecteuler.net/thread=250 - the best forum on the subject (note: you have to submit the correct solution first)
Code in various languages:
Python github.com/nayuki/Project-Euler-solutions/blob/master/python/p250.py (written by Nayuki)
Python github.com/steve98654/ProjectEuler/blob/master/250.py
C++ github.com/roosephu/project-euler/blob/master/250.cpp (written by Yuping Luo)
C++ github.com/smacke/project-euler/blob/master/cpp/250.cpp (written by Stephen Macke)
C github.com/LaurentMazare/ProjectEuler/blob/master/e250.c (written by Laurent Mazare)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p250.java (written by Nayuki)
Java github.com/thrap/project-euler/blob/master/src/Java/Problem250.java (written by Magnus Solheim Thrap)
Mathematica github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p250.mathematica (written by Nayuki)
Mathematica github.com/steve98654/ProjectEuler/blob/master/250.nb
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 | |
[new] | the flashing problem is the one I solved most recently |
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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.
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 249 - Prime Subset Sums | Reachable Numbers - problem 259 >> |