In [6]:
plotHelper()
from math import ceil as mC
from math import floor as mF
from math import log as mL
import matplotlib.pyplot as plt
import seaborn as sns
def getAllHeats(n):
"""
Creates all minimum heat level sets of N that can generate all heat levels from 1 to n.
Parameters:
none
Returns:
heats - List of all ordered N-tuples that satisfy obtaining all values of heats.
"""
def addOnHeats(N, heats, p) :
"""
Recursive function that adds one more viable heat level to all sets thus far.
Parameters:7
N - Final number of elements.
heats - Set of all heats.
p - Current number of elements in each heat set.
Returns:
heats - List of all ordered tuples thus far.
"""
#As long as the last spice is not being added...
if p < N-1 :
new_heats = []
for h in heats :
#The minimum heat level to be added must be at least the previous level.
minimum = h[-1]
if p >= 2 :
#If the previous two levels are the same, the minimum heat level to be added must be at least one more.
if h[-1] == h[-2] :
minimum += 1
#The maximum heat level is one more than the sum of all the previous heat levels.
maximum = sum(h) + 1
#Add new heats.
for j in range(minimum, maximum + 1) :
new_heats.append(h+[j])
#Keep adding to the set.
heats = addOnHeats(N, new_heats, p+1)
#Last level to be added.
else :
new_heats = []
for h in heats :
#The sum of all heat levels thus far must be at least half.
total = sum(h)
if total >= mF(n/2) :
#Add the remaining. It could be out of order. It could be a repeat.
h = tuple(sorted(h + [n-total]))
new_heats.append(h)
heats = list(set(new_heats))
return(heats)
#The minimum number of heat levels is the floor of log2 n + 1.
N = mF(mL(n, 2))+1
#All sets of N must start with 1.
heats = [[1]]
#Start recursion.
heats = addOnHeats(N, heats, 1)
return(N, sorted(heats))
len(getAllHeats(10)[1])
3
len(getAllHeats(100)[1])
114
def plotHelper() :
sns.set()
#Plot.
fig = plt.figure(figsize = (12, 6))
ax = fig.add_subplot(xlim = (1, 256))
#Title.
ax.set_title("How Many Sets?", fontsize = 24)
#x-axis.
ax.set_xlabel("Maximum Heat Level", fontsize = 18)
ax.set_xticks([2**x for x in range(2, 9)])
#y-axis.
ax.set_ylabel("Possible Sets to Generate", fontsize = 18)
ax.set_yscale("log")
# ax.set_yticks([2**(2*y) for y in range(8)])
ax.tick_params(axis = 'both', labelsize = 16)
#Data
n = list(range(1, 256))
heats = []
N = []
for i in n :
data = getAllHeats(i)
heats.append(len(data[1]))
N.append(data[0])
scatter = ax.scatter(x = n,
y = heats,
c = N,
cmap = 'viridis')
h, l = scatter.legend_elements()
l = ['1', '2', '3', '4', '5', '6', '7', '8']
legend = ax.legend(h, l,
title = "Size of Set",
title_fontsize = 18,
fontsize = '16',
loc = 'upper left',
ncol = 2)
ax.add_artist(legend);
fig.savefig('2025.11.28EC.png', bbox_inches = "tight");
plotHelper()
print(len(getAllHeats(256)[1]), len(getAllHeats(257)[1]))
462664 465297
getAllHeats(10)[1]
[(1, 1, 3, 5), (1, 2, 2, 5), (1, 2, 3, 4)]
getAllHeats(100)[1]
[(1, 2, 3, 6, 13, 25, 50), (1, 2, 3, 6, 13, 26, 49), (1, 2, 3, 7, 12, 25, 50), (1, 2, 3, 7, 12, 26, 49), (1, 2, 3, 7, 13, 24, 50), (1, 2, 3, 7, 13, 25, 49), (1, 2, 3, 7, 13, 26, 48), (1, 2, 3, 7, 13, 27, 47), (1, 2, 3, 7, 14, 23, 50), (1, 2, 3, 7, 14, 24, 49), (1, 2, 3, 7, 14, 25, 48), (1, 2, 3, 7, 14, 26, 47), (1, 2, 3, 7, 14, 27, 46), (1, 2, 3, 7, 14, 28, 45), (1, 2, 4, 5, 13, 25, 50), (1, 2, 4, 5, 13, 26, 49), (1, 2, 4, 6, 12, 25, 50), (1, 2, 4, 6, 12, 26, 49), (1, 2, 4, 6, 13, 24, 50), (1, 2, 4, 6, 13, 25, 49), (1, 2, 4, 6, 13, 26, 48), (1, 2, 4, 6, 13, 27, 47), (1, 2, 4, 6, 14, 23, 50), (1, 2, 4, 6, 14, 24, 49), (1, 2, 4, 6, 14, 25, 48), (1, 2, 4, 6, 14, 26, 47), (1, 2, 4, 6, 14, 27, 46), (1, 2, 4, 6, 14, 28, 45), (1, 2, 4, 7, 11, 25, 50), (1, 2, 4, 7, 11, 26, 49), (1, 2, 4, 7, 12, 24, 50), (1, 2, 4, 7, 12, 25, 49), (1, 2, 4, 7, 12, 26, 48), (1, 2, 4, 7, 12, 27, 47), (1, 2, 4, 7, 13, 23, 50), (1, 2, 4, 7, 13, 24, 49), (1, 2, 4, 7, 13, 25, 48), (1, 2, 4, 7, 13, 26, 47), (1, 2, 4, 7, 13, 27, 46), (1, 2, 4, 7, 13, 28, 45), (1, 2, 4, 7, 14, 22, 50), (1, 2, 4, 7, 14, 23, 49), (1, 2, 4, 7, 14, 24, 48), (1, 2, 4, 7, 14, 25, 47), (1, 2, 4, 7, 14, 26, 46), (1, 2, 4, 7, 14, 27, 45), (1, 2, 4, 7, 14, 28, 44), (1, 2, 4, 7, 14, 29, 43), (1, 2, 4, 7, 15, 21, 50), (1, 2, 4, 7, 15, 22, 49), (1, 2, 4, 7, 15, 23, 48), (1, 2, 4, 7, 15, 24, 47), (1, 2, 4, 7, 15, 25, 46), (1, 2, 4, 7, 15, 26, 45), (1, 2, 4, 7, 15, 27, 44), (1, 2, 4, 7, 15, 28, 43), (1, 2, 4, 7, 15, 29, 42), (1, 2, 4, 7, 15, 30, 41), (1, 2, 4, 8, 10, 25, 50), (1, 2, 4, 8, 10, 26, 49), (1, 2, 4, 8, 11, 24, 50), (1, 2, 4, 8, 11, 25, 49), (1, 2, 4, 8, 11, 26, 48), (1, 2, 4, 8, 11, 27, 47), (1, 2, 4, 8, 12, 23, 50), (1, 2, 4, 8, 12, 24, 49), (1, 2, 4, 8, 12, 25, 48), (1, 2, 4, 8, 12, 26, 47), (1, 2, 4, 8, 12, 27, 46), (1, 2, 4, 8, 12, 28, 45), (1, 2, 4, 8, 13, 22, 50), (1, 2, 4, 8, 13, 23, 49), (1, 2, 4, 8, 13, 24, 48), (1, 2, 4, 8, 13, 25, 47), (1, 2, 4, 8, 13, 26, 46), (1, 2, 4, 8, 13, 27, 45), (1, 2, 4, 8, 13, 28, 44), (1, 2, 4, 8, 13, 29, 43), (1, 2, 4, 8, 14, 21, 50), (1, 2, 4, 8, 14, 22, 49), (1, 2, 4, 8, 14, 23, 48), (1, 2, 4, 8, 14, 24, 47), (1, 2, 4, 8, 14, 25, 46), (1, 2, 4, 8, 14, 26, 45), (1, 2, 4, 8, 14, 27, 44), (1, 2, 4, 8, 14, 28, 43), (1, 2, 4, 8, 14, 29, 42), (1, 2, 4, 8, 14, 30, 41), (1, 2, 4, 8, 15, 20, 50), (1, 2, 4, 8, 15, 21, 49), (1, 2, 4, 8, 15, 22, 48), (1, 2, 4, 8, 15, 23, 47), (1, 2, 4, 8, 15, 24, 46), (1, 2, 4, 8, 15, 25, 45), (1, 2, 4, 8, 15, 26, 44), (1, 2, 4, 8, 15, 27, 43), (1, 2, 4, 8, 15, 28, 42), (1, 2, 4, 8, 15, 29, 41), (1, 2, 4, 8, 15, 30, 40), (1, 2, 4, 8, 15, 31, 39), (1, 2, 4, 8, 16, 19, 50), (1, 2, 4, 8, 16, 20, 49), (1, 2, 4, 8, 16, 21, 48), (1, 2, 4, 8, 16, 22, 47), (1, 2, 4, 8, 16, 23, 46), (1, 2, 4, 8, 16, 24, 45), (1, 2, 4, 8, 16, 25, 44), (1, 2, 4, 8, 16, 26, 43), (1, 2, 4, 8, 16, 27, 42), (1, 2, 4, 8, 16, 28, 41), (1, 2, 4, 8, 16, 29, 40), (1, 2, 4, 8, 16, 30, 39), (1, 2, 4, 8, 16, 31, 38), (1, 2, 4, 8, 16, 32, 37)]