Percolation/Bond percolation

From Rosetta Code
Task
Percolation/Bond percolation
You are encouraged to solve this task according to the task description, using any language you may know.

Percolation Simulation
This is a simulation of aspects of mathematical percolation theory.

For other percolation simulations, see Category:Percolation Simulations, or:
1D finite grid simulation
Mean run density
2D finite grid simulations

Site percolation | Bond percolation | Mean cluster density

Given an rectangular array of cells numbered , assume is horizontal and is downwards. Each is bounded by (horizontal) walls and ; (vertical) walls and

Assume that the probability of any wall being present is a constant where

Except for the outer horizontal walls at and which are always present.

The task

Simulate pouring a fluid onto the top surface () where the fluid will enter any empty cell it is adjacent to if there is no wall between where it currently is and the cell on the other side of the (missing) wall.

The fluid does not move beyond the horizontal constraints of the grid.

The fluid may move “up” within the confines of the grid of cells. If the fluid reaches a bottom cell that has a missing bottom wall then the fluid can be said to 'drip' out the bottom at that point.

Given repeat the percolation times to estimate the proportion of times that the fluid can percolate to the bottom for any given .

Show how the probability of percolating through the random grid changes with going from to in increments and with the number of repetitions to estimate the fraction at any given as .

Use an grid of cells for all cases.

Optionally depict fluid successfully percolating through a grid graphically.

Show all output on this page.

C[edit]

#include <stdio.h>
#include <stdlib.h>
#include <string.h>
 
// cell states
#define FILL 1
#define RWALL 2 // right wall
#define BWALL 4 // bottom wall
 
typedef unsigned int c_t;
 
c_t *cells, *start, *end;
int m, n;
 
void make_grid(double p, int x, int y)
{
int i, j, thresh = RAND_MAX * p;
m = x, n = y;
 
// Allocate two addition rows to avoid checking bounds.
// Bottom row is also required by drippage
start = realloc(start, m * (n + 2) * sizeof(c_t));
cells = start + m;
 
for (i = 0; i < m; i++)
start[i] = BWALL | RWALL;
 
for (i = 0, end = cells; i < y; i++) {
for (j = x; --j; )
*end++ = (rand() < thresh ? BWALL : 0)
|(rand() < thresh ? RWALL : 0);
*end++ = RWALL | (rand() < thresh ? BWALL: 0);
}
memset(end, 0, sizeof(c_t) * m);
}
 
void show_grid(void)
{
int i, j;
 
for (j = 0; j < m; j++) printf("+--");
puts("+");
 
for (i = 0; i <= n; i++) {
putchar(i == n ? ' ' : '|');
for (j = 0; j < m; j++) {
printf((cells[i*m + j] & FILL) ? "[]" : " ");
putchar((cells[i*m + j] & RWALL) ? '|' : ' ');
}
putchar('\n');
 
if (i == n) return;
 
for (j = 0; j < m; j++)
printf((cells[i*m + j] & BWALL) ? "+--" : "+ ");
puts("+");
}
}
 
int fill(c_t *p)
{
if ((*p & FILL)) return 0;
*p |= FILL;
if (p >= end) return 1; // success: reached bottom row
 
return ( !(p[ 0] & BWALL) && fill(p + m) ) ||
( !(p[ 0] & RWALL) && fill(p + 1) ) ||
( !(p[-1] & RWALL) && fill(p - 1) ) ||
( !(p[-m] & BWALL) && fill(p - m) );
}
 
int percolate(void)
{
int i;
for (i = 0; i < m && !fill(cells + i); i++);
 
return i < m;
}
 
int main(void)
{
make_grid(.5, 10, 10);
percolate();
show_grid();
 
int cnt, i, p;
 
puts("\nrunning 10x10 grids 10000 times for each p:");
for (p = 1; p < 10; p++) {
for (cnt = i = 0; i < 10000; i++) {
make_grid(p / 10., 10, 10);
cnt += percolate();
//show_grid(); // don't
}
printf("p = %3g: %.4f\n", p / 10., (double)cnt / i);
}
 
free(start);
return 0;
}
Output:
+--+--+--+--+--+--+--+--+--+--+
|[]|[] []|[] [] [] [] []      |
+  +  +  +--+--+--+--+  +  +--+
|[]|[]|[]|  |         [] []|  |
+  +--+  +--+--+--+  +--+  +  +
|[] [] [] []|        |   []|  |
+--+  +--+--+  +--+  +--+  +--+
|  |[]|     |  |        |[]   |
+--+--+  +  +  +  +  +--+  +  +
|     |  |     |         []   |
+--+  +  +--+  +--+--+  +  +--+
|  |     |     |[] [] [] []|  |
+  +  +  +--+  +  +--+--+--+--+
|  |  |     |   []   |  |  |  |
+--+  +--+--+--+  +  +  +--+  +
|  |  |  |  |  |[]|           |
+--+  +  +  +  +  +--+  +  +  +
|  |  |  |   [] []|  |  |  |  |
+--+  +--+--+  +--+  +  +  +  +
|  |     |   []|           |  |
+--+  +--+--+  +  +--+--+  +  +
             []                

running 10x10 grids 10000 times for each p:
p = 0.1: 1.0000
p = 0.2: 1.0000
p = 0.3: 0.9958
p = 0.4: 0.9123
p = 0.5: 0.5014
p = 0.6: 0.0791
p = 0.7: 0.0037
p = 0.8: 0.0000
p = 0.9: 0.0000

C++[edit]

Translation of: D
#include <cstdlib>
#include <cstring>
#include <iostream>
#include <string>
 
using namespace std;
 
class Grid {
public:
Grid(const double p, const int x, const int y) : m(x), n(y) {
const int thresh = static_cast<int>(RAND_MAX * p);
 
// Allocate two addition rows to avoid checking bounds.
// Bottom row is also required by drippage
start = new cell[m * (n + 2)];
cells = start + m;
for (auto i = 0; i < m; i++) start[i] = RBWALL;
end = cells;
for (auto i = 0; i < y; i++) {
for (auto j = x; --j;)
*end++ = (rand() < thresh ? BWALL : 0) | (rand() < thresh ? RWALL : 0);
*end++ = RWALL | (rand() < thresh ? BWALL : 0);
}
memset(end, 0u, sizeof(cell) * m);
}
 
~Grid() {
delete[] start;
cells = 0;
start = 0;
end = 0;
}
 
int percolate() const {
auto i = 0;
for (; i < m && !fill(cells + i); i++);
return i < m;
}
 
void show() const {
for (auto j = 0; j < m; j++)
cout << ("+-");
cout << '+' << endl;
 
for (auto i = 0; i <= n; i++) {
cout << (i == n ? ' ' : '|');
for (auto j = 0; j < m; j++) {
cout << ((cells[i * m + j] & FILL) ? "#" : " ");
cout << ((cells[i * m + j] & RWALL) ? '|' : ' ');
}
cout << endl;
 
if (i == n) return;
 
for (auto j = 0; j < m; j++)
cout << ((cells[i * m + j] & BWALL) ? "+-" : "+ ");
cout << '+' << endl;
}
}
 
private:
enum cell_state {
FILL = 1 << 0,
RWALL = 1 << 1, // right wall
BWALL = 1 << 2, // bottom wall
RBWALL = RWALL | BWALL // right/bottom wall
};
 
typedef unsigned int cell;
 
bool fill(cell* p) const {
if ((*p & FILL)) return false;
*p |= FILL;
if (p >= end) return true; // success: reached bottom row
 
return (!(p[0] & BWALL) && fill(p + m)) || (!(p[0] & RWALL) && fill(p + 1))
||(!(p[-1] & RWALL) && fill(p - 1)) || (!(p[-m] & BWALL) && fill(p - m));
}
 
cell* cells;
cell* start;
cell* end;
const int m;
const int n;
};
 
int main() {
const auto M = 10, N = 10;
const Grid grid(.5, M, N);
grid.percolate();
grid.show();
 
const auto C = 10000;
cout << endl << "running " << M << "x" << N << " grids " << C << " times for each p:" << endl;
for (auto p = 1; p < M; p++) {
auto cnt = 0, i = 0;
for (; i < C; i++)
cnt += Grid(p / static_cast<double>(M), M, N).percolate();
cout << "p = " << p / static_cast<double>(M) << ": " << static_cast<double>(cnt) / i << endl;
}
 
return EXIT_SUCCESS;
}

D[edit]

Translation of: C
import std.stdio, std.random, std.array, std.range, std.algorithm;
 
struct Grid {
// Not enforced by runtime and type system:
// a Cell must contain only the flags bits.
alias Cell = uint;
 
enum : Cell { // Cell states (bit flags).
empty = 0,
filled = 1,
rightWall = 2,
bottomWall = 4
}
 
const size_t nc, nr;
Cell[] cells;
 
this(in size_t nRows, in size_t nCols) pure nothrow {
nr = nRows;
nc = nCols;
 
// Allocate two addition rows to avoid checking bounds.
// Bottom row is also required by drippage.
cells = new Cell[nc * (nr + 2)];
}
 
void initialize(in double prob, ref Xorshift rng) {
cells[0 .. nc] = bottomWall | rightWall; // First row.
 
uint pos = nc;
foreach (immutable r; 1 .. nr + 1) {
foreach (immutable c; 1 .. nc)
cells[pos++] = (uniform01 < prob ?bottomWall : empty) |
(uniform01 < prob ? rightWall : empty);
cells[pos++] = rightWall |
(uniform01 < prob ? bottomWall : empty);
}
 
cells[$ - nc .. $] = empty; // Last row.
}
 
bool percolate() pure nothrow @nogc {
bool fill(in size_t i) pure nothrow @nogc {
if (cells[i] & filled)
return false;
 
cells[i] |= filled;
 
if (i >= cells.length - nc)
return true; // Success: reached bottom row.
 
return (!(cells[i] & bottomWall) && fill(i + nc)) ||
(!(cells[i] & rightWall) && fill(i + 1)) ||
(!(cells[i - 1] & rightWall) && fill(i - 1)) ||
(!(cells[i - nc] & bottomWall) && fill(i - nc));
}
 
return iota(nc, nc + nc).any!fill;
}
 
void show() const {
writeln("+-".replicate(nc), '+');
 
foreach (immutable r; 1 .. nr + 2) {
write(r == nr + 1 ? ' ' : '|');
foreach (immutable c; 0 .. nc) {
immutable cell = cells[r * nc + c];
write((cell & filled) ? (r <= nr ? '#' : 'X') : ' ');
write((cell & rightWall) ? '|' : ' ');
}
writeln;
 
if (r == nr + 1)
return;
 
foreach (immutable c; 0 .. nc)
write((cells[r * nc + c] & bottomWall) ? "+-" : "+ ");
'+'.writeln;
}
}
}
 
void main() {
enum uint nr = 10, nc = 10; // N. rows and columns of the grid.
enum uint nTries = 10_000; // N. simulations for each probability.
enum uint nStepsProb = 10; // N. steps of probability.
 
auto rng = Xorshift(2);
auto g = Grid(nr, nc);
g.initialize(0.5, rng);
g.percolate;
g.show;
 
writefln("\nRunning %dx%d grids %d times for each p:",
nr, nc, nTries);
foreach (immutable p; 0 .. nStepsProb) {
immutable probability = p / double(nStepsProb);
uint nPercolated = 0;
foreach (immutable i; 0 .. nTries) {
g.initialize(probability, rng);
nPercolated += g.percolate;
}
writefln("p = %0.2f: %.4f",
probability, nPercolated / double(nTries));
}
}
Output:
+-+-+-+-+-+-+-+-+-+-+
|#|#|#|#|     | |   |
+ +-+-+ +-+-+-+ +-+-+
|#| |  #  | | |   | |
+ +-+-+ + +-+-+ + +-+
|#|# #|#|   | |     |
+ +-+ + +-+ + + +-+ +
|#|# #|#|   | |   | |
+-+ + + + +-+-+-+-+-+
|# # # # #  | |   | |
+ + + + + + + +-+ +-+
|#|# # #|# # #  |   |
+-+ + + +-+-+ + + + +
| |#|# #| | |#      |
+-+-+-+-+ +-+ +-+-+-+
| |   |    # #|     |
+-+-+-+ +-+ +-+-+-+ +
| | |      # # #    |
+ + +-+ +-+-+-+ +-+ +
|     |   |   |#    |
+ +-+ + + + +-+ + + +
               X     

Running 10x10 grids 10000 times for each p:
p = 0.00: 1.0000
p = 0.10: 1.0000
p = 0.20: 1.0000
p = 0.30: 0.9973
p = 0.40: 0.9177
p = 0.50: 0.5050
p = 0.60: 0.0880
p = 0.70: 0.0035
p = 0.80: 0.0001
p = 0.90: 0.0000

With LDC2 compiler this code runs in 0.26 seconds (almost two times faster than the C entry).

Go[edit]

Translation of: C
package main
 
import (
"bytes"
"fmt"
"math/rand"
"time"
)
 
func main() {
const (
m, n = 10, 10
t = 1000
minp, maxp, Δp = 0.1, 0.99, 0.1
)
 
// Purposely don't seed for a repeatable example grid:
g := NewGrid(.5, m, n)
g.Percolate()
fmt.Println(g)
 
rand.Seed(time.Now().UnixNano()) // could pick a better seed
for p := float64(minp); p < maxp; p += Δp {
count := 0
for i := 0; i < t; i++ {
g := NewGrid(p, m, n)
if g.Percolate() {
count++
}
}
fmt.Printf("p=%.2f, %.3f\n", p, float64(count)/t)
}
}
 
type cell struct {
full bool
right, down bool // true if open to the right (x+1) or down (y+1)
}
 
type grid struct {
cell [][]cell // row first, i.e. [y][x]
}
 
func NewGrid(p float64, xsize, ysize int) *grid {
g := &grid{cell: make([][]cell, ysize)}
for y := range g.cell {
g.cell[y] = make([]cell, xsize)
for x := 0; x < xsize-1; x++ {
if rand.Float64() > p {
g.cell[y][x].right = true
}
if rand.Float64() > p {
g.cell[y][x].down = true
}
}
if rand.Float64() > p {
g.cell[y][xsize-1].down = true
}
}
return g
}
 
var (
full = map[bool]string{false: " ", true: "**"}
hopen = map[bool]string{false: "--", true: " "}
vopen = map[bool]string{false: "|", true: " "}
)
 
func (g *grid) String() string {
var buf bytes.Buffer
// Don't really need to call Grow but it helps avoid multiple
// reallocations if the size is large.
buf.Grow((len(g.cell) + 1) * len(g.cell[0]) * 7)
 
for _ = range g.cell[0] {
buf.WriteString("+")
buf.WriteString(hopen[false])
}
buf.WriteString("+\n")
for y := range g.cell {
buf.WriteString(vopen[false])
for x := range g.cell[y] {
buf.WriteString(full[g.cell[y][x].full])
buf.WriteString(vopen[g.cell[y][x].right])
}
buf.WriteByte('\n')
for x := range g.cell[y] {
buf.WriteString("+")
buf.WriteString(hopen[g.cell[y][x].down])
}
buf.WriteString("+\n")
}
ly := len(g.cell) - 1
for x := range g.cell[ly] {
buf.WriteByte(' ')
buf.WriteString(full[g.cell[ly][x].down && g.cell[ly][x].full])
}
return buf.String()
}
 
func (g *grid) Percolate() bool {
for x := range g.cell[0] {
if g.fill(x, 0) {
return true
}
}
return false
}
 
func (g *grid) fill(x, y int) bool {
if y >= len(g.cell) {
return true // Out the bottom
}
if g.cell[y][x].full {
return false // Allready filled
}
g.cell[y][x].full = true
 
if g.cell[y][x].down && g.fill(x, y+1) {
return true
}
if g.cell[y][x].right && g.fill(x+1, y) {
return true
}
if x > 0 && g.cell[y][x-1].right && g.fill(x-1, y) {
return true
}
if y > 0 && g.cell[y-1][x].down && g.fill(x, y-1) {
return true
}
return false
}
Output:
+--+--+--+--+--+--+--+--+--+--+
|** ** **|  |  |     |  |  |  |
+  +--+  +--+--+  +--+--+--+  +
|**|  |** **|  |     |     |  |
+--+  +--+  +  +  +--+  +  +--+
|     |   **|  |              |
+--+  +--+  +--+--+--+--+--+--+
|     |   ** **|        |     |
+--+  +  +--+  +  +--+  +--+  +
|           |** ** **|     |  |
+  +  +--+  +--+  +  +--+  +--+
|  |  |  |   ** ** ** **|  |  |
+  +--+--+  +  +--+--+  +--+--+
|  |** ** **|**|**|  |** ** **|
+  +  +  +  +  +  +--+  +--+  +
|** **|**|** ** **|  |** ** **|
+  +  +--+--+--+--+  +  +--+  +
|**|** ** **|     |  |**|  |**|
+  +--+--+--+  +  +--+--+--+--+
|**               |  |  |  |  |
+  +  +  +  +--+--+  +--+--+  +
 **                           
p=0.10, 1.000
p=0.20, 1.000
p=0.30, 0.998
p=0.40, 0.915
p=0.50, 0.502
p=0.60, 0.081
p=0.70, 0.002
p=0.80, 0.000
p=0.90, 0.000

Perl 6[edit]

Works with: Rakudo version 2017.02

Starts "filling" from the top left. Fluid flow favours directions in Down, Left, Right, Up order. I interpreted p to be porosity, so small p mean low permeability, large p means high permeability.

my @bond;
my $grid = 10;
my $geom = $grid - 1;
my $water = '▒';
 
enum Direction <DeadEnd Up Right Down Left>;
 
say 'Sample percolation at .6';
percolate .6;
.join.say for @bond;
say "\n";
 
my $tests = 100;
say "Doing $tests trials at each porosity:";
for .1, .2 ... 1 -> $p {
printf "p = %0.1f: %0.2f\n", $p, (sum percolate($p) xx $tests) / $tests
}
 
sub percolate ( $prob ) {
generate $prob;
my @stack;
my $current = [1;0];
$current.&fill;
 
loop {
if my $dir = direction( $current ) {
@stack.push: $current;
$current = move $dir, $current
}
else {
return 0 unless @stack;
$current = @stack.pop
}
return 1 if $current[1] == +@bond - 1
}
 
sub direction( [$x, $y] ) {
( Down if @bond[$y + 1][$x].contains: ' ' ) ||
( Left if @bond[$y][$x - 1].contains: ' ' ) ||
( Right if @bond[$y][$x + 1].contains: ' ' ) ||
( Up if @bond[$y - 1][$x].defined && @bond[$y - 1][$x].contains: ' ' ) ||
DeadEnd
}
 
sub move ( $dir, @cur ) {
my ( $x, $y ) = @cur;
given $dir {
when Up { [$x,--$y].&fill xx 2 }
when Down { [$x,++$y].&fill xx 2 }
when Left { [--$x,$y].&fill xx 2 }
when Right { [++$x,$y].&fill xx 2 }
}
[$x, $y]
}
 
sub fill ( [$x, $y] ) { @bond[$y;$x].=subst(' ', $water, :g) }
}
 
sub generate ( $prob = .5 ) {
@bond = ();
my $sp = ' ';
append @bond, [flat '│', ($sp, ' ') xx $geom, $sp, '│'],
[flat '├', (h(), '┬') xx $geom, h(), '┤'];
append @bond, [flat '│', ($sp, v()) xx $geom, $sp, '│'],
[flat '├', (h(), '┼') xx $geom, h(), '┤'] for ^$geom;
append @bond, [flat '│', ($sp, v()) xx $geom, $sp, '│'],
[flat '├', (h(), '┴') xx $geom, h(), '┤'],
[flat '│', ($sp, ' ') xx $geom, $sp, '│'];
 
sub h () { rand < $prob ?? $sp !! '───' }
sub v () { rand < $prob ?? ' ' !! '│' }
}
Output:
Sample percolation at .6
│▒▒▒                                    │
├▒▒▒┬   ┬───┬   ┬   ┬   ┬   ┬   ┬───┬   ┤
│▒▒▒▒▒▒▒                │   │           │
├───┼▒▒▒┼   ┼   ┼   ┼   ┼   ┼───┼   ┼   ┤
│▒▒▒▒▒▒▒▒▒▒▒│   │   │   │   │   │       │
├▒▒▒┼───┼▒▒▒┼   ┼───┼   ┼───┼   ┼   ┼   ┤
│▒▒▒│▒▒▒▒▒▒▒▒▒▒▒                │   │   │
├▒▒▒┼───┼───┼▒▒▒┼   ┼   ┼───┼   ┼   ┼   ┤
│▒▒▒│        ▒▒▒│   │   │           │   │
├───┼   ┼   ┼▒▒▒┼───┼   ┼   ┼   ┼   ┼───┤
│           │▒▒▒    │                   │
├   ┼───┼   ┼▒▒▒┼───┼───┼───┼───┼   ┼───┤
│           │▒▒▒│                       │
├───┼   ┼───┼▒▒▒┼───┼───┼───┼───┼   ┼───┤
│▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒        │       │       │
├▒▒▒┼▒▒▒┼───┼▒▒▒┼───┼   ┼───┼   ┼   ┼   ┤
│▒▒▒│▒▒▒▒▒▒▒│▒▒▒▒▒▒▒│                   │
├▒▒▒┼───┼───┼───┼───┼───┼   ┼   ┼   ┼   ┤
│▒▒▒▒▒▒▒    │       │   │               │
├▒▒▒┼▒▒▒┼───┼───┼   ┼───┼───┼   ┼   ┼   ┤
│▒▒▒│▒▒▒    │               │           │
├───┴▒▒▒┴   ┴   ┴   ┴───┴   ┴   ┴   ┴───┤
│    ▒▒▒                                │


Doing 100 trials at each porosity:
p = 0.1: 0.00
p = 0.2: 0.00
p = 0.3: 0.00
p = 0.4: 0.05
p = 0.5: 0.42
p = 0.6: 0.92
p = 0.7: 1.00
p = 0.8: 1.00
p = 0.9: 1.00
p = 1.0: 1.00

Python[edit]

from collections import namedtuple
from random import random
from pprint import pprint as pp
 
Grid = namedtuple('Grid', 'cell, hwall, vwall')
 
M, N, t = 10, 10, 100
 
class PercolatedException(Exception): pass
 
HVF = [(' .', ' _'), (':', '|'), (' ', '#')] # Horiz, vert, fill chars
 
def newgrid(p):
hwall = [[int(random() < p) for m in range(M)]
for n in range(N+1)]
vwall = [[(1 if m in (0, M) else int(random() < p)) for m in range(M+1)]
for n in range(N)]
cell = [[0 for m in range(M)]
for n in range(N)]
return Grid(cell, hwall, vwall)
 
def pgrid(grid, percolated=None):
cell, hwall, vwall = grid
h, v, f = HVF
for n in range(N):
print(' ' + ''.join(h[hwall[n][m]] for m in range(M)))
print('%i) ' % (n % 10) + ''.join(v[vwall[n][m]] + f[cell[n][m] if m < M else 0]
for m in range(M+1))[:-1])
n = N
print(' ' + ''.join(h[hwall[n][m]] for m in range(M)))
if percolated:
where = percolated.args[0][0]
print('!) ' + ' ' * where + ' ' + f[1])
 
def pour_on_top(grid):
cell, hwall, vwall = grid
n = 0
try:
for m in range(M):
if not hwall[n][m]:
flood_fill(m, n, cell, hwall, vwall)
except PercolatedException as ex:
return ex
return None
 
 
def flood_fill(m, n, cell, hwall, vwall):
# fill cell
cell[n][m] = 1
# bottom
if n < N - 1 and not hwall[n + 1][m] and not cell[n+1][m]:
flood_fill(m, n+1, cell, hwall, vwall)
# THE bottom
elif n == N - 1 and not hwall[n + 1][m]:
raise PercolatedException((m, n+1))
# left
if m and not vwall[n][m] and not cell[n][m - 1]:
flood_fill(m-1, n, cell, hwall, vwall)
# right
if m < M - 1 and not vwall[n][m + 1] and not cell[n][m + 1]:
flood_fill(m+1, n, cell, hwall, vwall)
# top
if n and not hwall[n][m] and not cell[n-1][m]:
flood_fill(m, n-1, cell, hwall, vwall)
 
if __name__ == '__main__':
sample_printed = False
pcount = {}
for p10 in range(11):
p = (10 - p10) / 10.0 # count down so sample print is interesting
pcount[p] = 0
for tries in range(t):
grid = newgrid(p)
percolated = pour_on_top(grid)
if percolated:
pcount[p] += 1
if not sample_printed:
print('\nSample percolating %i x %i grid' % (M, N))
pgrid(grid, percolated)
sample_printed = True
print('\n p: Fraction of %i tries that percolate through' % t )
 
pp({p:c/float(t) for p, c in pcount.items()})
Output:

In the Ascii art, cells are either a space or a hash and are surrounded by either '_', '|' for intact walls and '.' and ':' for missing (leaky) walls.

The bottom-most line starting '!)' shows where the fluid can drip out from. (The percolation stops when one route through the bottom is found).

Sample percolating 10 x 10 grid
     _ _ . _ . _ _ . _ _
0)  | |#:#:#|#| | :#| | |
     _ _ . _ _ _ . . _ _
1)  | | |#:#| | | |#| : |
     _ _ _ . _ . . . . _
2)  | | |#:#| : | |#: | |
     _ _ _ _ . . _ . . .
3)  | : : | | | : |#: | |
     _ _ . _ . . _ . _ _
4)  | : : : | | | |#: : |
     _ _ _ . _ _ _ . . _
5)  | : | | : | | :#| | |
     _ _ . . _ _ _ . _ .
6)  | : | | : | |#:#:#| |
     _ . _ _ . _ _ _ . .
7)  | : | : | : | | |#: |
     _ _ _ . . _ _ . . _
8)  | | : | | | |#:#:#: |
     _ _ _ . . . . _ _ .
9)  | : : | : : :#: | : |
     . _ . _ . . . . _ _
!)               #

 p: Fraction of 100 tries that percolate through
{0.0: 1.0,
 0.1: 1.0,
 0.2: 1.0,
 0.3: 1.0,
 0.4: 0.9,
 0.5: 0.47,
 0.6: 0.06,
 0.7: 0.0,
 0.8: 0.0,
 0.9: 0.0,
 1.0: 0.0}

Note the abrupt cut-off in percolation at around p = 0.5 which is to be expected.

Racket[edit]

#lang racket
 
(define has-left-wall? (lambda (x) (bitwise-bit-set? x 0)))
(define has-right-wall? (lambda (x) (bitwise-bit-set? x 1)))
(define has-top-wall? (lambda (x) (bitwise-bit-set? x 2)))
(define has-bottom-wall? (lambda (x) (bitwise-bit-set? x 3)))
(define has-fluid? (lambda (x) (bitwise-bit-set? x 4)))
 
(define (walls->cell l? r? t? b?)
(+ (if l? 1 0) (if r? 2 0) (if t? 4 0) (if b? 8 0)))
 
(define (bonded-percol-grid M N p)
(define rv (make-vector (* M N)))
(for* ((idx (in-range (* M N))))
(define left-wall?
(or (zero? (modulo idx M))
(has-right-wall? (vector-ref rv (sub1 idx)))))
(define right-wall?
(or (= (modulo idx M) (sub1 M))
(< (random) p)))
(define top-wall?
(if (< idx M) (< (random) p)
(has-bottom-wall? (vector-ref rv (- idx M)))))
(define bottom-wall? (< (random) p))
(define cell-value
(walls->cell left-wall? right-wall? top-wall? bottom-wall?))
(vector-set! rv idx cell-value))
rv)
 
(define (display-percol-grid M . vs)
(define N (/ (vector-length (car vs)) M))
(define-syntax-rule (tab-eol m)
(when (= m (sub1 M)) (printf "\t")))
(for ((n N))
(for* ((v vs) (m M))
(when (zero? m) (printf "+"))
(printf
(match (vector-ref v (+ (* n M) m))
((? has-top-wall?) "-+")
((? has-fluid?) "#+")
(else ".+")))
(tab-eol m))
(newline)
(for* ((v vs) (m M))
(when (zero? m) (printf "|"))
(printf
(match (vector-ref v (+ (* n M) m))
((and (? has-fluid?) (? has-right-wall?)) "#|")
((? has-right-wall?) ".|")
((? has-fluid?) "##")
(else "..")))
(tab-eol m))
(newline))
(for* ((v vs) (m M))
(when (zero? m) (printf "+"))
(printf
(match (vector-ref v (+ (* (sub1 M) M) m))
((? has-bottom-wall?) "-+")
((? has-fluid?) "#+")
(else ".+")))
(tab-eol m))
(newline))
 
(define (find-bonded-grid-t/b-path M v)
(define N (/ (vector-length v) M))
 
(define (flood-cell idx)
(cond
[(= (quotient idx M) N) #t] ; wootiments!
[(has-fluid? (vector-ref v idx)) #f] ; been here
[else (define cell (vector-ref v idx))
(vector-set! v idx (bitwise-ior cell 16))
(or (and (not (has-bottom-wall? cell)) (flood-cell (+ idx M)))
(and (not (has-left-wall? cell)) (flood-cell (- idx 1)))
(and (not (has-right-wall? cell)) (flood-cell (+ idx 1)))
(and (not (has-top-wall? cell))
(>= idx M) ; not top row
(flood-cell (- idx M))))]))
 
(for/first ((m (in-range M))
#:unless (has-top-wall? (vector-ref v m))
#:when (flood-cell m)) #t))
 
(define t (make-parameter 1000))
(define (experiment p)
(/ (for*/sum ((sample (in-range (t)))
(v (in-value (bonded-percol-grid 10 10 p)))
#:when (find-bonded-grid-t/b-path 10 v)) 1)
(t)))
 
(define (main)
(for ((tenths (in-range 0 (add1 10))))
(define p (/ tenths 10))
(define e (experiment p))
(printf "proportion of grids that percolate p=~a : ~a (~a)~%"
p e (real->decimal-string e 5))))
 
(module+ test
(define (make/display/flood/display-bonded-grid M N p attempts (atmpt 1))
(define v (bonded-percol-grid M N p))
(define v+ (vector-copy v))
(cond [(or (find-bonded-grid-t/b-path M v+) (= attempts 0))
(define v* (vector-copy v+))
(define (flood-bonded-grid)
(when (find-bonded-grid-t/b-path M v*)
(flood-bonded-grid)))
(flood-bonded-grid)
(display-percol-grid M v v+ v*)
(printf "After ~a attempt(s)~%~%" atmpt)]
[else
(make/display/flood/display-bonded-grid
M N p (sub1 attempts) (add1 atmpt))]))
 
(make/display/flood/display-bonded-grid 10 10 0 20)
(make/display/flood/display-bonded-grid 10 10 .25 20)
(make/display/flood/display-bonded-grid 10 10 .50 20)
(make/display/flood/display-bonded-grid 10 10 .75 20000))
Output:
Welcome to DrRacket, version 5.3.5 [3m].
Language: racket [custom]; memory limit: 1024 MB.
+.+.+.+.+.+.+.+.+.+.+	+#+.+.+.+.+.+.+.+.+.+	+#+#+#+#+#+#+#+#+#+#+	
|...................|	|##.................|	|###################|	
+.+.+.+.+.+.+.+.+.+.+	+#+.+.+.+.+.+.+.+.+.+	+#+#+#+#+#+#+#+#+#+#+	
|...................|	|##.................|	|###################|	
+.+.+.+.+.+.+.+.+.+.+	+#+.+.+.+.+.+.+.+.+.+	+#+#+#+#+#+#+#+#+#+#+	
|...................|	|##.................|	|###################|	
+.+.+.+.+.+.+.+.+.+.+	+#+.+.+.+.+.+.+.+.+.+	+#+#+#+#+#+#+#+#+#+#+	
|...................|	|##.................|	|###################|	
+.+.+.+.+.+.+.+.+.+.+	+#+.+.+.+.+.+.+.+.+.+	+#+#+#+#+#+#+#+#+#+#+	
|...................|	|##.................|	|###################|	
+.+.+.+.+.+.+.+.+.+.+	+#+.+.+.+.+.+.+.+.+.+	+#+#+#+#+#+#+#+#+#+#+	
|...................|	|##.................|	|###################|	
+.+.+.+.+.+.+.+.+.+.+	+#+.+.+.+.+.+.+.+.+.+	+#+#+#+#+#+#+#+#+#+#+	
|...................|	|##.................|	|###################|	
+.+.+.+.+.+.+.+.+.+.+	+#+.+.+.+.+.+.+.+.+.+	+#+#+#+#+#+#+#+#+#+#+	
|...................|	|##.................|	|###################|	
+.+.+.+.+.+.+.+.+.+.+	+#+.+.+.+.+.+.+.+.+.+	+#+#+#+#+#+#+#+#+#+#+	
|...................|	|##.................|	|###################|	
+.+.+.+.+.+.+.+.+.+.+	+#+.+.+.+.+.+.+.+.+.+	+#+#+#+#+#+#+#+#+#+#+	
|...................|	|##.................|	|###################|	
+.+.+.+.+.+.+.+.+.+.+	+#+.+.+.+.+.+.+.+.+.+	+#+#+#+#+#+#+#+#+#+#+	
After 1 attempt(s)

+.+-+-+.+.+.+-+.+.+.+	+#+-+-+.+.+.+-+.+.+.+	+#+-+-+#+#+#+-+#+#+#+	
|...................|	|##.................|	|##..###############|	
+.+-+.+-+.+.+-+-+-+.+	+#+-+.+-+.+.+-+-+-+.+	+#+-+#+-+#+#+-+-+-+#+	
|.................|.|	|##...............|.|	|##..##..####.....|#|	
+.+-+.+.+.+.+-+.+.+.+	+#+-+.+.+.+.+-+.+.+.+	+#+-+#+.+#+#+-+.+.+#+	
|.............|.....|	|##...........|.....|	|######..#####|....#|	
+.+.+.+.+.+.+.+.+.+.+	+#+.+.+.+.+.+.+.+.+.+	+#+#+#+.+#+#+#+.+.+#+	
|.....|...|.|.......|	|##...|...|.|.......|	|#####|..#|#|##....#|	
+.+.+.+.+.+.+.+-+-+.+	+#+.+.+.+.+.+.+-+-+.+	+#+#+#+#+#+#+#+-+-+#+	
|.|.............|...|	|#|.............|...|	|#|############.|..#|	
+.+-+-+.+-+.+.+.+.+.+	+#+-+-+.+-+.+.+.+.+.+	+#+-+-+#+-+#+#+.+.+#+	
|...................|	|##.................|	|##....##..####....#|	
+.+.+-+.+.+.+.+-+-+.+	+#+.+-+.+.+.+.+-+-+.+	+#+.+-+#+.+#+#+-+-+#+	
|...|...|...........|	|##.|...|...........|	|##.|###|..####..###|	
+.+.+.+-+.+.+.+.+.+.+	+#+#+.+-+.+.+.+.+.+.+	+#+#+#+-+.+#+#+.+#+#+	
|...|...|.........|.|	|###|...|.........|.|	|###|##.|..####..#|#|	
+-+.+.+-+-+.+.+.+.+-+	+-+#+.+-+-+.+.+.+.+-+	+-+#+#+-+-+#+#+.+#+-+	
|.....|.........|...|	|..##.|.........|...|	|..###|....####.|###|	
+.+.+.+.+.+.+.+.+.+.+	+.+#+.+.+.+.+.+.+.+.+	+.+#+#+.+.+#+#+#+#+#+	
|.........|.......|.|	|..##.....|.......|.|	|..####...|#######|#|	
+.+.+.+-+.+.+-+.+-+.+	+.+#+.+-+.+.+-+.+-+.+	+.+#+#+-+.+#+-+#+-+#+	
After 1 attempt(s)

+.+.+.+.+-+-+.+-+.+.+	+#+#+#+#+-+-+.+-+.+.+	+#+#+#+#+-+-+#+-+#+#+	
|.........|.|.|...|.|	|########.|.|.|...|.|	|########.|.|#|###|#|	
+.+-+-+.+-+-+-+.+.+-+	+#+-+-+#+-+-+-+.+.+-+	+#+-+-+#+-+-+-+#+#+-+	
|...|...|...|.|.|.|.|	|###|..#|...|.|.|.|.|	|###|..#|...|.|#|#|.|	
+-+-+.+.+.+.+-+.+-+.+	+-+-+.+#+#+.+-+.+-+.+	+-+-+.+#+#+.+-+#+-+.+	
|.|.|.|...|.|.|.|...|	|.|.|.|###|.|.|.|...|	|.|.|.|###|.|.|#|...|	
+.+-+.+-+.+.+.+-+.+-+	+.+-+.+-+#+.+.+-+.+-+	+.+-+.+-+#+.+.+-+.+-+	
|.|...|...|.|.....|.|	|.|...|###|.|.....|.|	|.|...|###|.|.....|.|	
+.+-+.+.+.+-+-+.+.+.+	+.+-+.+#+#+-+-+.+.+.+	+.+-+.+#+#+-+-+.+.+.+	
|.|...|.|.....|.....|	|.|...|#|####.|.....|	|.|...|#|####.|.....|	
+-+.+-+.+-+.+-+.+-+-+	+-+.+-+#+-+#+-+#+-+-+	+-+.+-+#+-+#+-+#+-+-+	
|.|.|.....|.....|...|	|.|.|#####|#####|...|	|.|.|#####|#####|...|	
+-+-+.+.+.+.+-+.+-+-+	+-+-+#+#+#+#+-+#+-+-+	+-+-+#+#+#+#+-+#+-+-+	
|...|.|.....|.......|	|...|#|#####|..##...|	|...|#|#####|..##...|	
+-+-+-+-+-+-+-+.+-+-+	+-+-+-+-+-+-+-+#+-+-+	+-+-+-+-+-+-+-+#+-+-+	
|.|...|.|.|.......|.|	|.|...|.|.|######.|.|	|.|...|.|.|######.|.|	
+.+-+-+-+.+.+-+.+.+.+	+.+-+-+-+.+#+-+#+.+.+	+.+-+-+-+.+#+-+#+.+.+	
|.|...|.......|.|.|.|	|.|...|....##.|#|.|.|	|.|...|....##.|#|.|.|	
+.+.+-+.+.+.+-+-+-+-+	+.+.+-+.+.+#+-+-+-+-+	+.+.+-+.+.+#+-+-+-+-+	
|.|.........|.....|.|	|.|........#|.....|.|	|.|........#|.....|.|	
+-+.+-+-+-+.+.+.+-+.+	+-+.+-+-+-+#+.+.+-+.+	+-+.+-+-+-+#+.+.+-+.+	
After 2 attempt(s)

+-+-+-+-+-+-+.+-+-+.+	+-+-+-+-+-+-+#+-+-+.+	+-+-+-+-+-+-+#+-+-+#+	
|.|.|...|.|.|.|.|...|	|.|.|...|.|.|#|.|...|	|.|.|...|.|.|#|.|###|	
+-+-+-+-+-+-+.+-+-+-+	+-+-+-+-+-+-+#+-+-+-+	+-+-+-+-+-+-+#+-+-+-+	
|.|.|.|...|.|...|.|.|	|.|.|.|...|.|##.|.|.|	|.|.|.|...|.|##.|.|.|	
+.+.+.+.+.+-+.+-+-+.+	+.+.+.+.+.+-+#+-+-+.+	+.+.+.+.+.+-+#+-+-+.+	
|.|.|.|.|...|.|...|.|	|.|.|.|.|...|#|...|.|	|.|.|.|.|...|#|...|.|	
+.+-+.+-+-+-+.+-+.+.+	+.+-+.+-+-+-+#+-+.+.+	+.+-+.+-+-+-+#+-+.+.+	
|...|...|.|.|...|.|.|	|...|...|.|.|###|.|.|	|...|...|.|.|###|.|.|	
+-+-+-+-+-+-+-+.+-+-+	+-+-+-+-+-+-+-+#+-+-+	+-+-+-+-+-+-+-+#+-+-+	
|.|.......|.....|.|.|	|.|.......|#####|.|.|	|.|.......|#####|.|.|	
+.+-+-+-+.+.+-+-+-+-+	+.+-+-+-+.+#+-+-+-+-+	+.+-+-+-+.+#+-+-+-+-+	
|.|.|.|.|.|.|.|.....|	|.|.|.|.|.|#|.|.....|	|.|.|.|.|.|#|.|.....|	
+-+-+-+-+-+.+.+-+.+-+	+-+-+-+-+-+#+.+-+.+-+	+-+-+-+-+-+#+.+-+.+-+	
|...|.|.|.|.|.|.|.|.|	|...|.|.|.|#|.|.|.|.|	|...|.|.|.|#|.|.|.|.|	
+.+.+.+-+-+.+.+-+.+-+	+.+.+.+-+-+#+#+-+.+-+	+.+.+.+-+-+#+#+-+.+-+	
|.|.|.|.|.|...|.|...|	|.|.|.|.|.|###|.|...|	|.|.|.|.|.|###|.|...|	
+-+-+-+-+-+-+.+.+.+-+	+-+-+-+-+-+-+#+.+.+-+	+-+-+-+-+-+-+#+.+.+-+	
|.|.|.|.|.|...|...|.|	|.|.|.|.|.|###|...|.|	|.|.|.|.|.|###|...|.|	
+-+-+.+-+-+.+-+-+-+.+	+-+-+.+-+-+#+-+-+-+.+	+-+-+.+-+-+#+-+-+-+.+	
|.|.|.|...|...|.|...|	|.|.|.|...|###|.|...|	|.|.|.|...|###|.|...|	
+-+-+.+-+.+-+.+.+.+-+	+-+-+.+-+.+-+#+.+.+-+	+-+-+.+-+.+-+#+.+.+-+	
After 4611 attempt(s)

> (main)
proportion of grids that percolate p=0 : 1 (1.00000)
proportion of grids that percolate p=1/10 : 1 (1.00000)
proportion of grids that percolate p=1/5 : 1 (1.00000)
proportion of grids that percolate p=3/10 : 199/200 (0.99500)
proportion of grids that percolate p=2/5 : 179/200 (0.89500)
proportion of grids that percolate p=1/2 : 451/1000 (0.45100)
proportion of grids that percolate p=3/5 : 29/500 (0.05800)
proportion of grids that percolate p=7/10 : 1/1000 (0.00100)
proportion of grids that percolate p=4/5 : 0 (0.00000)
proportion of grids that percolate p=9/10 : 0 (0.00000)
proportion of grids that percolate p=1 : 0 (0.00000)

Tcl[edit]

Works with: Tcl version 8.6
Translation of: Python
package require Tcl 8.6
 
# Structure the bond percolation system as a class
oo::class create BondPercolation {
variable hwall vwall cells M N
constructor {width height probability} {
set M $height
set N $width
for {set i 0} {$i <= $height} {incr i} {
for {set j 0;set walls {}} {$j < $width} {incr j} {
lappend walls [expr {rand() < $probability}]
}
lappend hwall $walls
}
for {set i 0} {$i <= $height} {incr i} {
for {set j 0;set walls {}} {$j <= $width} {incr j} {
lappend walls [expr {$j==0 || $j==$width || rand() < $probability}]
}
lappend vwall $walls
}
set cells [lrepeat $height [lrepeat $width 0]]
}
 
method print {{percolated ""}} {
set nw [string length $M]
set grid $cells
if {$percolated ne ""} {
lappend grid [lrepeat $N 0]
lset grid end $percolated 1
}
foreach hws $hwall vws [lrange $vwall 0 end-1] r $grid {
incr row
puts -nonewline [string repeat " " [expr {$nw+2}]]
foreach w $hws {
puts -nonewline [if {$w} {subst "+-"} {subst "+ "}]
}
puts "+"
puts -nonewline [format "%-*s" [expr {$nw+2}] [expr {
$row>$M ? $percolated eq "" ? " " : ">" : "$row)"
}]]
foreach v $vws c $r {
puts -nonewline [if {$v==1} {subst "|"} {subst " "}]
puts -nonewline [if {$c==1} {subst "#"} {subst " "}]
}
puts ""
}
}
 
method percolate {} {
try {
for {set i 0} {$i < $N} {incr i} {
if {![lindex $hwall 0 $i]} {
my FloodFill $i 0
}
}
return ""
} trap PERCOLATED n {
return $n
}
}
method FloodFill {x y} {
# fill cell
lset cells $y $x 1
# bottom
if {![lindex $hwall [expr {$y+1}] $x]} {
if {$y == $N-1} {
# THE bottom
throw PERCOLATED $x
}
if {$y < $N-1 && ![lindex $cells [expr {$y+1}] $x]} {
my FloodFill $x [expr {$y+1}]
}
}
# left
if {![lindex $vwall $y $x] && ![lindex $cells $y [expr {$x-1}]]} {
my FloodFill [expr {$x-1}] $y
}
# right
if {![lindex $vwall $y [expr {$x+1}]] && ![lindex $cells $y [expr {$x+1}]]} {
my FloodFill [expr {$x+1}] $y
}
# top
if {$y>0 && ![lindex $hwall $y $x] && ![lindex $cells [expr {$y-1}] $x]} {
my FloodFill $x [expr {$y-1}]
}
}
}
 
# Demonstrate one run
puts "Sample percolation, 10x10 p=0.5"
BondPercolation create bp 10 10 0.5
bp print [bp percolate]
bp destroy
puts ""
 
# Collect some aggregate statistics
apply {{} {
puts "Percentage of tries that percolate, varying p"
set tries 100
for {set pint 0} {$pint <= 10} {incr pint} {
set p [expr {$pint * 0.1}]
set tot 0
for {set i 0} {$i < $tries} {incr i} {
set bp [BondPercolation new 10 10 $p]
if {[$bp percolate] ne ""} {
incr tot
}
$bp destroy
}
puts [format "p=%.2f: %2.1f%%" $p [expr {$tot*100./$tries}]]
}
}}
Output:
Sample percolation, 10x10 p=0.5
    + + +-+-+-+ +-+ +-+ +
1)  |#  |   |   |   |   | 
    + +-+ + + +-+ + + +-+
2)  |#|       | |     | | 
    + + +-+ +-+ +-+ + +-+
3)  |# # #|# # #| | |   | 
    + +-+ + +-+ +-+ +-+ +
4)  |#|# # #| |#  |     | 
    +-+ + + +-+ +-+-+ +-+
5)  |# # # #| |#  |   | | 
    +-+-+-+-+ + + + +-+-+
6)  | |     | |#|   |   | 
    +-+-+-+-+-+ + +-+-+ +
7)  | | | |   |#      | | 
    + +-+ +-+-+ +-+ +-+ +
8)  |       |  #    |   | 
    + +-+-+ +-+ + + + + +
9)  |          #        | 
    + + +-+-+ + +-+-+ + +
10) |   | |    #  | |   | 
    + + + + + + +-+ +-+ +
>              #        

Percentage of tries that percolate, varying p
p=0.00: 100.0%
p=0.10: 100.0%
p=0.20: 100.0%
p=0.30: 100.0%
p=0.40: 86.0%
p=0.50: 50.0%
p=0.60: 6.0%
p=0.70: 0.0%
p=0.80: 0.0%
p=0.90: 0.0%
p=1.00: 0.0%

zkl[edit]

Translation of: C
// cell states
const FILLED=1; // and odd
const RWALL =2; // right wall
const BWALL =4; // bottom wall
fcn P(p,wall){ (0.0).random(1)<p and wall or 0 }
 
fcn makeGrid(m,n,p){
// Allocate two addition rows to avoid checking bounds.
// Bottom row is also required by drippage
grid:=Data(m*(n+2));
do(m){ grid.write(BWALL + RWALL); } // grid is topped with walls
do(n){
do(m-1){ grid.write( P(p,BWALL) + P(p,RWALL) ) }
grid.write(RWALL + P(p,BWALL)); // right border is all right wall, as is left border
}
do(m){ grid.write(0); } // for drips off the bottom of grid
grid
}
fcn show(grid,m,n){ n+=1;
println("+--"*m,"+");
foreach i in ([1..n]){ y:=i*m;
print(i==n and " " or "|"); // bottom row is special, otherwise always have left wall
foreach j in (m){ c:=grid[y + j];
print(c.bitAnd(FILLED) and "**" or " ", c.bitAnd(RWALL)and"|"or" ");
}
println();
 
if(i==n) return(); // nothing under the bottom row
 
foreach j in (m){ print((grid[y + j].bitAnd(BWALL)) and "+--" or "+ "); }
println("+");
}
}
fcn fill(grid,x,m){
if(grid[x].isOdd) return(False); // aka .bitAnd(FILLED) aka already been here
grid[x]+=FILLED;
if(x+m>=grid.len()) return(True); // success: reached bottom row
return(( not grid[x] .bitAnd(BWALL) and fill(grid,x + m,m) ) or // down
( not grid[x] .bitAnd(RWALL) and fill(grid,x + 1,m) ) or // right
( not grid[x - 1].bitAnd(RWALL) and fill(grid,x - 1,m) ) or // left
( not grid[x - m].bitAnd(BWALL) and fill(grid,x - m,m) )); // up
}
fcn percolate(grid,m){
i:=0; while(i<m and not fill(grid,i+m,m)){ i+=1; } // pour juice on top row
return(i<m); // percolated through the grid?
}
grid:=makeGrid(10,10,0.40);
println("Did liquid percolate: ",percolate(grid,10)); show(grid,10,10);
 
println("Running 10,000 tests for each case:");
foreach p in ([0.0 .. 1.0, 0.1]){
cnt:=0.0; do(10000){ cnt+=percolate(makeGrid(10,10,p),10); }
"p=%.1f:  %.4f".fmt(p, cnt/10000).println();
}
Output:
Did liquid percolate: True
+--+--+--+--+--+--+--+--+--+--+
|** **      |              |  |
+--+  +--+--+  +  +  +  +  +  +
|   **|  |        |        |  |
+  +  +  +--+  +--+--+  +--+--+
|   ** **      |              |
+--+--+  +  +  +  +--+  +  +--+
|     |**|  |        |        |
+  +  +  +  +--+  +  +--+  +  +
|      ** **|  |** **|     |  |
+  +--+--+  +--+  +  +--+  +  +
|     |  |**|  |**|** **      |
+  +  +  +  +--+  +--+  +  +  +
|     |  |** ** ** **|**      |
+--+--+--+--+  +--+--+  +--+--+
|  |     |** **|      **   |  |
+  +  +--+  +  +  +  +  +--+--+
|        |** **|     |**|  |  |
+  +--+  +--+--+--+--+  +  +  +
|              |  |   **|     |
+  +  +  +  +  +  +--+  +  +  +
                      **       
Running 10,000 tests for each case:
p=0.0:  1.0000
p=0.1:  1.0000
p=0.2:  1.0000
p=0.3:  0.9978
p=0.4:  0.9163
p=0.5:  0.5017
p=0.6:  0.0890
p=0.7:  0.0033
p=0.8:  0.0000
p=0.9:  0.0000
p=1.0:  0.0000