<< problem 300 - Protein folding | Multiples with small digits - problem 303 >> |
Problem 301: Nim
(see projecteuler.net/problem=301)
Nim is a game played with heaps of stones, where two players take it in turn to remove any number of stones from any heap until no stones remain.
We'll consider the three-heap normal-play version of Nim, which works as follows:
- At the start of the game there are three heaps of stones.
- On his turn the player removes any positive number of stones from any single heap.
- The first player unable to move (because no stones remain) loses.
that you may look up or attempt to deduce for yourself that returns:
- zero if, with perfect strategy, the player about to move will eventually lose; or
- non-zero if, with perfect strategy, the player about to move will eventually win.
- current player moves to (1,2,1)
- opponent moves to (1,0,1)
- current player moves to (0,0,1)
- opponent moves to (0,0,0), and so wins.
My Algorithm
This Wikipedia page told me that the "Nim Sum" is the XOR-result of all heaps: en.wikipedia.org/wiki/Nim
Alternative Approaches
Often I enter a few results into my search engine of choice - this time I was surprised to see that
the results for 2^1, 2^2, 2^3, ... 2^n are the (n+2)-th Fibonacci numbers.
Someone with more mathematical insight could probably seen that before writing the code - but I can't.
It's possible to write a Dynamic Programming solution, too:
when n contains no two consecutive set bits then the game is lost.
Note
I reversed the loop to iterate from 10^30 to 0 because then my compiler can produce code that is about 10% faster.
Interactive test
You can submit your own input to my program and it will be instantly processed at my server:
This is equivalent toecho 10 | ./301
Output:
Note: the original problem's input 30
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. Or just jump to my GitHub repository.
#include <iostream>
// see en.wikipedia.org/wiki/Nim
unsigned int evaluate(unsigned int n1, unsigned int n2, unsigned int n3)
{
return n1 ^ n2 ^ n3;
}
int main()
{
// search up to n = 2^exponent
unsigned int exponent = 30;
std::cin >> exponent;
unsigned int lost = 0;
// 2^30 => about 1 billion values
for (unsigned int n = 1U << exponent; n > 0; n--)
{
auto score = evaluate(n, 2*n, 3*n);
if (score == 0)
lost++;
}
std::cout << lost << std::endl;
return 0;
}
This solution contains 4 empty lines, 3 comments and 1 preprocessor command.
Benchmark
The correct solution to the original Project Euler problem was found in 0.9 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
July 4, 2017 submitted solution
July 4, 2017 added comments
Difficulty
Project Euler ranks this problem at 15% (out of 100%).
Links
projecteuler.net/thread=301 - the best forum on the subject (note: you have to submit the correct solution first)
Code in various languages:
Python github.com/LaurentMazare/ProjectEuler/blob/master/e301.py (written by Laurent Mazare)
Python github.com/Meng-Gen/ProjectEuler/blob/master/301.py (written by Meng-Gen Tsai)
Python github.com/nayuki/Project-Euler-solutions/blob/master/python/p301.py (written by Nayuki)
Python github.com/steve98654/ProjectEuler/blob/master/301.py
C++ github.com/steve98654/ProjectEuler/blob/master/301.cpp
C github.com/shlomif/project-euler/blob/master/project-euler/301/euler_301.c (written by Shlomi Fish)
Java github.com/dcrousso/ProjectEuler/blob/master/PE301.java (written by Devin Rousso)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p301.java (written by Nayuki)
Java github.com/thrap/project-euler/blob/master/src/Java/Problem301.java (written by Magnus Solheim Thrap)
Go github.com/frrad/project-euler/blob/master/golang/Problem301.go (written by Frederick Robinson)
Mathematica github.com/nayuki/Project-Euler-solutions/blob/master/mathematica/p301.mathematica (written by Nayuki)
Haskell github.com/nayuki/Project-Euler-solutions/blob/master/haskell/p301.hs (written by Nayuki)
Perl github.com/gustafe/projecteuler/blob/master/301-Nim.pl (written by Gustaf Erikson)
Sage github.com/roosephu/project-euler/blob/master/301.sage (written by Yuping Luo)
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 |
I stopped working on Project Euler problems around the time they released 617.
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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 300 - Protein folding | Multiples with small digits - problem 303 >> |