<< problem 348 - Sum of a square and a cube Prime generating integers - problem 357 >>

# Problem 349: Langton's ant

An ant moves on a regular grid of squares that are coloured either black or white.
The ant is always oriented in one of the cardinal directions (left, right, up or down) and moves from square to adjacent square according to the following rules:
- if it is on a black square, it flips the color of the square to white, rotates 90 degrees counterclockwise and moves forward one square.
- if it is on a white square, it flips the color of the square to black, rotates 90 degrees clockwise and moves forward one square.

Starting with a grid that is entirely white, how many squares are black after 10^18 moves of the ant?

# My Algorithm

I had no idea how to solve this problem until I read the Wikipedia page about Langton's ant: en.wikipedia.org/wiki/Langton's_ant
One thing was especially interesting: the initial movement of the ant may be chaotic, but after a while a certain pattern develops which repeats after 104 steps.

My program simulates the movement of the ant on a 64x64 grid (starting in the middle).
It counts the number of black squares and every 104 steps (= 1 cycle) it computes the delta compared to the number of black squares 104 steps ago.
As soon as I observe the same difference over at least 10 cycles I can easily find out how many cycles are needed for 10^18 steps (minus the steps already taken).

To simplify my computation I don't check the difference at the start of a cycle but after the 40th steps of a cycle because 10^18 mod 104 = 40.

## Note

The recurring pattern appears after about 10000 steps.

# 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>

int main()
{
auto limit = 1000000000000000000LL;
std::cin >> limit;

// colors encoded with a single bit
const bool White = false;
const bool Black = true;

// 128x128 squares (nice power of two with sufficient squares for the first 10000 steps)
const unsigned int Size = 128;

// 2D grid stored as a 1D array: pos = y * Size + x;
std::vector<bool> grid(Size * Size, White);

// initial position of the ant
unsigned int x = Size / 2;
unsigned int y = Size / 2;

// delta movement when ant moves Up, Left, Down, Right
const short DeltaX[] = {  0, +1,  0, -1 };
const short DeltaY[] = { +1,  0, -1,  0 };
// direction 0..3 (see DeltaX and DeltaY)
short direction = 0; // any direction is fine

// a pattern with cycle length 104 emerges after a while (see https://en.wikipedia.org/wiki/Langton%27s_ant )
const auto Cycle     = 104;
const auto Remainder = limit % Cycle; // = 40

// number of black squares
auto count = 0LL;
// and the value of "count" 104 steps ago
auto last = count;

// keep track of the deltas (which are "count-last")
std::vector<int> lastDeltas = { 0 };
// assume a "steady state" if the most recent 10 cycles have the same deltas
const unsigned int StopIfSameDeltas = 10;

// number of steps taken
auto steps = 0;
while (steps < limit)
{
// check every 104 steps whether the "steady state" was achieved
if (steps % Cycle == Remainder)
{
// change of Black squares
auto diff = int(count - last);
lastDeltas.push_back(diff);
last = count;

// need a few samples ...
if (lastDeltas.size() >= StopIfSameDeltas)
{
bool samesame = true;
// always the same diff during the most recent cycles ?
for (auto scan = lastDeltas.rbegin(); scan != lastDeltas.rbegin() + StopIfSameDeltas; scan++)
if (*scan != diff)
{
// nope, found a different difference :-(
samesame = false;
break;
}

// yes, entered "steady state"
if (samesame)
{
// determine number of cycles left
auto remainingCycles = (limit - steps) / Cycle;
// and multiply by 12
count += remainingCycles * diff;
// stop searching
break;
}
}
}

// ant moves one step
auto pos = y * Size + x;
if (grid[pos] == White)
{
// flip square
grid[pos] = Black;
count++;

// counter-clockwise
direction = (direction + 1) % 4;
}
else // Black
{
// flip square
grid[pos] = White;
count--;

// clockwise
direction = (direction + 4 - 1) % 4; // plus 4 avoids negative numbers
}

// "That's one small step for an ant, one giant leap for antkind"
x += DeltaX[direction];
y += DeltaY[direction];
steps++;
}

// display result
std::cout << count << std::endl;
return 0;
}


This solution contains 17 empty lines, 28 comments and 2 preprocessor commands.

# 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 the number of moves.

This is equivalent to
echo 10 | ./349

Output:

(please click 'Go !')

Note: the original problem's input 1000000000000000000 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)

# Benchmark

The correct solution to the original Project Euler problem was found in less than 0.01 seconds on a Intel® Core™ i7-2600K CPU @ 3.40GHz.
(compiled for x86_64 / Linux, GCC flags: -O3 -march=native -fno-exceptions -fno-rtti -std=c++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 19, 2017 submitted solution

# Difficulty

Project Euler ranks this problem at 35% (out of 100%).

# Heatmap

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The 239 solved problems (level 9) had an average difficulty of 29.1% at Project Euler and
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 << problem 348 - Sum of a square and a cube Prime generating integers - problem 357 >>
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