<< problem 300 - Protein folding Multiples with small digits - problem 303 >>

# Problem 301: Nim

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.
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:

• 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.
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:
• 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.
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.

       #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%).

# 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.

 << problem 300 - Protein folding Multiples with small digits - problem 303 >>
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