Partitioning Hands of Dice, Better

dice_part.py

#!/usr/bin/python3

# Starting over, a little cleaner and a little better
# Ian McDougall, March 2020

import math
import numpy as np
import matplotlib as mpl
import matplotlib.pyplot as plt

### Dice Functions

def decrement(X):
	if sum(X) == len(X):	#this is a kludge to make the while loop stop
		X.pop()
		X.append(0)
		return X
	X.sort(reverse=True)
	Y = X.pop()
	if Y > 1:
		Y -= 1
		X.append(Y)
	elif Y == 1:
		decrement(X)
		X.append(X[-1])
	return X

def karmarkar_karp(X):
	Y = list(X)
	while len(Y) > 1:
		Y.sort()
		diff = Y.pop() - Y.pop()
		Y.append(diff)
	return Y[0]

def count_perms(X):
	cts = [0]
	cts *= max(X)
	den = 1
	for i in X:	#think of this like a really bad hash function
		cts[i-1] += 1	#i-1 accounts for the zero-indexing
	for i in cts:
		den = den*math.factorial(i)
	num = math.factorial(len(X))
	return num/den

def mechanic(no_dice, no_sides):
	roll = [no_sides]
	roll *= no_dice
	dice = []
	while sum(roll) > (len(roll)-1):
		dice += [list(roll)]
		decrement(roll)
	return dice

### Exploratory Functions

def chance_diff(no_dice, no_sides):
	rolls = mechanic(no_dice, no_sides)
	rolls_diff = np.array([karmarkar_karp(hand) for hand in rolls], dtype=bool)
#	rolls_diff != 0	#this comparison seems equivalent to "dtype=bool", but causes numerical errors at scale
	rolls_perms = np.array([count_perms(hand) for hand in rolls])
	chance = sum( rolls_diff * rolls_perms / sum(rolls_perms))
	return chance

### Plotting Functions

def my_hexbin(ax, sums, diffs, odds):
	#todo: force integer tick labels
	xgrid = (max(sums) - min(sums))//2
	ygrid = (max(diffs) - min(diffs))//2
	ax.hexbin(sums, diffs, C=odds, gridsize=(xgrid, ygrid), cmap='Blues')
	ax.set_xlabel('Maximum Difference (Sum)')
	ax.set_ylabel('Minimum Difference')
	ax.set_frame_on(False)
	ax.tick_params(length=0)
	return ax

def my_hist(ax, sums, diffs, odds, dice):
	his = 0.5 * (sums + diffs)
	los = 0.5 * (sums - diffs)
	half = 0.5 * sums
	ax.hist(
		np.hstack((his, los)),
		np.arange(min(los)-0.5, max(his)+1.5, 1),
		density=1,
		rwidth=0.5,
		label=dice+', Partitioned',
		weights=np.hstack((odds,odds))
		)
	ax.hist(
		half,
		np.arange(min(half)-0.25, max(half)+0.75, 0.5),
		density=1,
		histtype='step',
		label=dice+', Halved',
		weights=odds
		)
	ax.set_xlabel('Individual Scores')
	ax.set_yticks([])
	ax.set_frame_on(False)
	ax.tick_params(length=0)
	ax.legend(frameon=False)
	return ax

### Testing with 7d6

m7d6 = mechanic(7,6)
m7d6_sum = np.array([sum(hand) for hand in m7d6])
m7d6_diff = np.array([karmarkar_karp(hand) for hand in m7d6])
m7d6_perms = np.array([count_perms(hand) for hand in m7d6])

w, h = mpl.figure.figaspect(0.29)
#aspect is from a hexagon (sqrt(3)/2) and the grid sizes
fig1, ax11 = plt.subplots(figsize=(w,h))
my_hexbin(ax11, m7d6_sum, m7d6_diff, m7d6_perms)
fig1.savefig('figure1.png', bbox_inches='tight')

fig2, ax21 = plt.subplots()
my_hist(ax21, m7d6_sum, m7d6_diff, m7d6_perms, '7d6')
fig2.savefig('figure2.png', bbox_inches='tight')

### Exploration

count = [3, 4, 5, 6, 7, 8]	#defining numbers of dice to roll & partition
sides = [4, 6, 8, 10, 12, 20]	#defining the common dice

chances = np.array([[chance_diff(a, b) for b in sides] for a in count])
fig3, ax31 = plt.subplots()
im = ax31.imshow(chances, cmap='binary', norm=mpl.colors.Normalize(0,1))
ax31.tick_params(top=True, bottom=False, labeltop=True, labelbottom=False, length=0)
ax31.set_xticks(np.arange(len(sides)))
ax31.set_xticklabels(['d{}'.format(sides[a]) for a in np.arange(len(sides))])
ax31.set_yticks(np.arange(len(count)))
ax31.set_yticklabels(count)
for i in range(len(count)):
	for j in range(len(sides)):
		text = ax31.text(j, i, '{:4.2f}'.format(chances[i, j]), ha='center', va='center', color='w')
for edge, spine in ax31.spines.items():
	spine.set_visible=False
plt.box(None)
fig3.savefig('figure3.png', bbox_inches='tight')

### Testing with 3d20

m3d20 = mechanic(3,20)
m3d20_sum = np.array([sum(hand) for hand in m3d20])
m3d20_diff = np.array([karmarkar_karp(hand) for hand in m3d20])
m3d20_perms = np.array([count_perms(hand) for hand in m3d20])

w, h = mpl.figure.figaspect(0.63)
fig4, ax41 = plt.subplots(figsize=(w,h))
my_hexbin(ax41, m3d20_sum, m3d20_diff, m3d20_perms)
fig4.savefig('figure4.png', bbox_inches='tight')

fig5, ax51 = plt.subplots()
my_hist(ax51, m3d20_sum, m3d20_diff, m3d20_perms, '3d20')
fig5.savefig('figure5.png', bbox_inches='tight')