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:

If (n1,n2,n3) indicates a Nim position consisting of heaps of size n1, n2 and n3 then there is a simple function X(n1,n2,n3)
that you may look up or attempt to deduce for yourself that returns:

For example X(1,2,3) = 0 because, no matter what the current player does, his opponent can respond with a move that leaves two heaps of equal size, at which point every move by the current player can be mirrored by his opponent until no stones remain; so the current player loses. To illustrate:
For how many positive integers n <= 2^30 does X(n,2n,3n) = 0 ?

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:

Input data (separated by spaces or newlines):
Note: Enter n which gives the upper limit 2^n.

This is equivalent to
echo 10 | ./301

Output:

(please click 'Go !')

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

15% 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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The 310 solved problems (that's level 12) had an average difficulty of 32.6% at Project Euler and
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 !!!

more about me can be found on my homepage, especially in my coding blog.
some names mentioned on this site may be trademarks of their respective owners.
thanks to the KaTeX team for their great typesetting library !