A Game of Numbers

[Cancer]

It takes a while to compute the complete distribution of 10d12 by brute force.

[Old Man]

26 minutes ago, Cancer said:

It takes a while to compute the complete distribution of 10d12 by brute force.

 

import random

# Set the number of dice and sides
num_dice = 10
num_sides = 12

# Roll the dice and record the results
results = [random.randint(1, num_sides) for i in range(num_dice)]

# Compute the probability distribution
dist = {}
for result in results:
    if result in dist:
        dist[result] += 1
    else:
        dist[result] = 1
for result in sorted(dist.keys()):
    print(f"{result}: {dist[result]/num_dice:.2%}")

[Cancer]

Of course.  I was doing it by exhaustion, deterministically.

[Old Man]

My bad.

 

from itertools import product

# Set the number of dice and sides
num_dice = 10
num_sides = 12

# Compute the probability of each possible outcome
outcomes = product(range(1, num_sides+1), repeat=num_dice)
dist = {}
for outcome in outcomes:
    total = sum(outcome)
    if total in dist:
        dist[total] += 1
    else:
        dist[total] = 1
for total in sorted(dist.keys()):
    print(f"{total}: {dist[total]/num_sides**num_dice:.6%}")

[Cancer]

Runtime: hours, with 6e+10 numbers to accumulate.

[Old Man]

24 minutes ago, Cancer said:

Runtime: hours, with 6e+10 numbers to accumulate.

 

We'll see.  This Macbook is over ten years old so it won't be quick.

[Old Man]

No idea how long this would have taken if my laptop hadn't slept for a few hours here and there.

 

10d12 start:
08:47:49

10: 0.000000%
11: 0.000000%
12: 0.000000%
13: 0.000000%
14: 0.000001%
15: 0.000003%
16: 0.000008%
17: 0.000018%
18: 0.000039%
19: 0.000079%
20: 0.000149%
21: 0.000271%
22: 0.000475%
23: 0.000803%
24: 0.001319%
25: 0.002108%
26: 0.003288%
27: 0.005014%
28: 0.007489%
29: 0.010970%
30: 0.015782%
31: 0.022322%
32: 0.031068%
33: 0.042588%
34: 0.057541%
35: 0.076679%
36: 0.100841%
37: 0.130946%
38: 0.167976%
39: 0.212955%
40: 0.266924%
41: 0.330900%
42: 0.405843%
43: 0.492605%
44: 0.591883%
45: 0.704165%
46: 0.829685%
47: 0.968366%
48: 1.119785%
49: 1.283138%
50: 1.457212%
51: 1.640383%
52: 1.830613%
53: 2.025481%
54: 2.222213%
55: 2.417747%
56: 2.608800%
57: 2.791954%
58: 2.963755%
59: 3.120819%
60: 3.259932%
61: 3.378163%
62: 3.472957%
63: 3.542225%
64: 3.584421%
65: 3.598594%
66: 3.584421%
67: 3.542225%
68: 3.472957%
69: 3.378163%
70: 3.259932%
71: 3.120819%
72: 2.963755%
73: 2.791954%
74: 2.608800%
75: 2.417747%
76: 2.222213%
77: 2.025481%
78: 1.830613%
79: 1.640383%
80: 1.457212%
81: 1.283138%
82: 1.119785%
83: 0.968366%
84: 0.829685%
85: 0.704165%
86: 0.591883%
87: 0.492605%
88: 0.405843%
89: 0.330900%
90: 0.266924%
91: 0.212955%
92: 0.167976%
93: 0.130946%
94: 0.100841%
95: 0.076679%
96: 0.057541%
97: 0.042588%
98: 0.031068%
99: 0.022322%
100: 0.015782%
101: 0.010970%
102: 0.007489%
103: 0.005014%
104: 0.003288%
105: 0.002108%
106: 0.001319%
107: 0.000803%
108: 0.000475%
109: 0.000271%
110: 0.000149%
111: 0.000079%
112: 0.000039%
113: 0.000018%
114: 0.000008%
115: 0.000003%
116: 0.000001%
117: 0.000000%
118: 0.000000%
119: 0.000000%
120: 0.000000%
10d12 stop:
20:44:22

[Old Man]

I forgot to mention that credit for writing the Python code should go to... ChatGPT.

[Cancer]

Maybe you can talk it into rewriting itself.

[Logan D. Hurricanes]

Today is World Maths Day! 

 

 

 

 

It is also Be Nasty Day, but I'm sure that's just a coincidence. 

[Cancer]

1287821 seems to be largest palindrome number in the first 100,000 primes.

[Bazza]

[Cancer]

That's something like a Sieve of Eratosthenes plotted in polar coordinates.  Interesting.

[Bazza]

Yes, I think it comes from the same source (ie I remember the Sieve of Eratosthenes mentioned from the DVD I just watching last night. (I got the image from the author's website). 

I'm sure you picked up that the circle is divided into 24 segments and noticed the Maltese Cross. 

I thought of posting it to the latest Superdraft thread it it is related to my academic school of qualitative geometry. More to come after the weekend.

[Cancer]

I suspect that if you plotted many more numbers on that pattern, the Maltese cross would become muted, perhaps fade away entirely, or become restricted to an inner core, once you had, say, 10000 primes in the display.  Other patterns might appear. 

 

Plotting it with 24 segments to the circle means that important sequences of composite numbers (multiples of 2, 3, 4, 6, and 12) appear as rays emanating from the center.  Carefully increasing the number of segments would do that for more such sequences.  The next important case is 60 segments around the circle, and then the multiples of 5 would make radial rays.  After that you need 420 segments so that the multiples of 7 make rays of their own as well. 

 

I hadn't seen this sort of format before, and I find myself thinking about what it would look like in a 3-dimensional spherical plot and how the best way to display such a thing would work.  You'd need to use zonal harmonics, I think, but there could well be other tricks that don't occur to me.

[Cancer]

Though I just noticed this morning an oddity: 1 is indicated as a prime (which technically it is not), and 2 and 3 are not indicated as  primes, which both most definitely are.  Those corrections mar the Maltese cross pattern, of course.

[Bazza]

Yes, agree with you. Will hopeful to reply tomorrow (later today) in full when it is not 12:56 am Monday. 

[Logan D. Hurricanes]

[Logan D. Hurricanes]

High school students make math discovery 2,000 years in the making

[Logan D. Hurricanes]

[Bazza]

*chuckle*

[Cancer]

Yeah, it should be 42, not zero.

[Logan D. Hurricanes]

[Logan D. Hurricanes]

[Cancer]

On the down side, I think that at least a third of the kids in my Solar System astronomy class this term can't read scientific notation.  They'd rather count zeros than see 2 ×10³⁰ kg.