結論：30只以上為佳，3的倍數
https://www.bilibili.com/video/BV17nYVekEWU/?vd_source=9b97428fe30142d2bb4f9bab6ef082a4
# Import necessary libraries
import numpy as np
import random
from collections import defaultdict
import matplotlib.pyplot as plt
# Function to initialize the game with a specified number of turtles with random colors
def initialize_game(n):
colors = list(range(10)) # Define ten possible colors for the turtles
initial_turtles = [random.choice(colors) for _ in range(n)] # Initialize turtles with random colors
return initial_turtles
# Function to draw a turtle from the pool; may modify the pool if the lucky color is drawn
def draw_turtle(turtle_pool, lucky_color):
if turtle_pool:
# Randomly select and remove a turtle from the pool
turtle = turtle_pool.pop(random.randint(0, len(turtle_pool) - 1))
# If the turtle's color matches the lucky color, add a new random turtle to the pool
if turtle == lucky_color:
turtle_pool.append(random.choice(range(10)))
return turtle
return None
# Function to fill a 3x3 grid with turtles from the pool
def fill_grid(turtle_pool, grid, lucky_color):
for i in range(3):
for j in range(3):
# Only place a new turtle in empty grid spaces
if grid[i][j] is None and turtle_pool:
grid[i][j] = draw_turtle(turtle_pool, lucky_color)
# Function to check for three in a row or column and update the grid accordingly
def check_and_update_grid(grid, turtle_pool, lucky_color):
to_remove = []
three_in_a_row_count = 0
two_in_a_row_count = 0
# Check for three in a row/column
for i in range(3):
if grid[i][0] is not None and grid[i][0] == grid[i][1] == grid[i][2]:
to_remove.extend([(i, 0), (i, 1), (i, 2)])
three_in_a_row_count += 1
for j in range(3):
if grid[0][j] is not None and grid[0][j] == grid[1][j] == grid[2][j]:
to_remove.extend([(0, j), (1, j), (2, j)])
three_in_a_row_count += 1
# Remove turtles from the grid
for i, j in to_remove:
grid[i][j] = None
# Refill turtle pool if three in a row were found
if to_remove:
for _ in range(5 * three_in_a_row_count):
turtle_pool.append(random.choice(range(10)))
# Check for two in a row/column and clear them
pairs = []
for i in range(3):
for j in range(3):
if grid[i][j] is not None:
for x in range(i, 3):
for y in range(3):
if (i != x or j != y) and grid[i][j] == grid[x][y]:
pairs.append((i, j, x, y))
# Remove pairs of turtles from the grid
if pairs:
i, j, x, y = pairs[0]
grid[i][j] = None
grid[x][y] = None
turtle_pool.append(random.choice(range(10)))
two_in_a_row_count += 1
# Check if all grid colors are different
colors_in_grid = set()
all_different = True
for i in range(3):
for j in range(3):
if grid[i][j] is not None:
if grid[i][j] in colors_in_grid:
all_different = False
colors_in_grid.add(grid[i][j])
else:
all_different = False
# Clear the grid if all colors are different and reward with more turtles
if all_different:
for i in range(3):
for j in range(3):
grid[i][j] = None
for _ in range(5):
turtle_pool.append(random.choice(range(10)))
return (three_in_a_row_count, two_in_a_row_count, all_different)
# Function to determine if there are any pairs of turtles left in the grid
def grid_has_pairs(grid):
"""Check if there are any removable groups of turtles in the grid"""
for i in range(3):
if grid[i][0] is not None and grid[i][0] == grid[i][1] == grid[i][2]:
return True
for j in range(3):
if grid[0][j] is not None and grid[0][j] == grid[1][j] == grid[2][j]:
return True
for i in range(3):
for j in range(3):
if grid[i][j] is not None:
for x in range(i, 3):
for y in range(3):
if (i != x or j != y) and grid[i][j] == grid[x][y]:
return True
return False
# Main function to simulate the gameplay
def play_game(n, lucky_color):
turtle_pool = initialize_game(n)
grid = [[None for _ in range(3)] for _ in range(3)]
total_draws = n
three_in_a_row_count = 0
two_in_a_row_count = 0
lucky_turtle_draws = 0
all_different_count = 0
# Continue game while there are moves left
while turtle_pool or grid_has_pairs(grid):
fill_grid(turtle_pool, grid, lucky_color)
# Check grid and update it while there are matches
while True:
three_count, two_count, all_different = check_and_update_grid(grid, turtle_pool, lucky_color)
if three_count == 0 and two_count == 0 and not all_different:
break
three_in_a_row_count += three_count
two_in_a_row_count += two_count
if all_different:
all_different_count += 1
# Check for and count lucky color draws
for row in grid:
for turtle in row:
if turtle == lucky_color:
lucky_turtle_draws += 1
# Calculate total number of draws
total_draws = n + lucky_turtle_draws + two_in_a_row_count + 5 * three_in_a_row_count + 5 * all_different_count
return total_draws
# Simulation function to run multiple trials and analyze results
def run_simulation(trials, n):
results = []
lucky_color = random.choice(range(10))
for _ in range(trials):
m = play_game(n, lucky_color)
results.append(m)
return np.array(results)
# Main function to conduct simulations and plot results
def main():
n_list=[]
m_mean_list=[]
m_variance_list=[]
m_mean_div_n_list=[]
trials = 10000
for n in range(1, 11, 1):
results = run_simulation(trials, n)
mean = np.mean(results)
variance = np.var(results)
distribution = dict(zip(*np.unique(results, return_counts=True)))
print(f'n = {n}:')
print(f' Mean = {mean}')
print(f' Variance = {variance}')
print(f' Probability Distribution = {distribution}\n')
n_list.append(n)
m_mean_list.append(mean)
m_variance_list.append(variance)
m_mean_div_n_list.append(mean/n)
plt.plot(n_list, m_mean_list)
plt.plot(n_list, m_variance_list)
plt.plot(n_list, m_mean_div_n_list)
# Check if the script is the main program and run it
if __name__ == "__main__":
main()