Exact distribution of beginner 3BV

[Tjips]

As some of you might already know Daniele Calafiore (Peppinos) and I have been trying to count the possible beginner boards, and we finally have the results of Daniele's program. Here is the normalized distribution of the 3BV:

Beg3BVDist.jpg (83.8 KiB) Viewed 6735 times

The average 3BV is 17.203. The most probable 3BV value is 16 and the total number of boards (ignoring symmetry) is 151473214816. The actual numbers are attached...

[dcalafiore]

Follwing a table with number , probability and frequency of every possible beginner 3bv.


3bv | number of boards | probability | there is a board with that 3bv every...

1 | 771160 | 5,09E-006 | 196.422,6
2 | 7463738 | 4,93E-005 | 20.294,6
3 | 37896050 | 2,50E-004 | 3.997,1
4 | 132535786 | 8,75E-004 | 1.142,9
5 | 357978312 | 2,36E-003 | 423,1
6 | 796360012 | 5,26E-003 | 190,2
7 | 1519742142 | 1,00E-002 | 99,7
8 | 2560164382 | 1,69E-002 | 59,2
9 | 3891189862 | 2,57E-002 | 38,9
10 | 5425642786 | 3,58E-002 | 27,9
11 | 7017180914 | 4,63E-002 | 21,6
12 | 8525476012 | 5,63E-002 | 17,8
13 | 9785712676 | 6,46E-002 | 15,5
14 | 10719728572 | 7,08E-002 | 14,1
15 | 11214156036 | 7,40E-002 | 13,5
16 | 11321990274 | 7,47E-002 | 13,4
17 | 11011690082 | 7,27E-002 | 13,8
18 | 10417203504 | 6,88E-002 | 14,5
19 | 9541989694 | 6,30E-002 | 15,9
20 | 8539586782 | 5,64E-002 | 17,7
21 | 7479084044 | 4,94E-002 | 20,3
22 | 6380228210 | 4,21E-002 | 23,7
23 | 5319278192 | 3,51E-002 | 28,5
24 | 4432378596 | 2,93E-002 | 34,2
25 | 3535773296 | 2,33E-002 | 42,8
26 | 2818658126 | 1,86E-002 | 53,7
27 | 2216273268 | 1,46E-002 | 68,3
28 | 1703144354 | 1,12E-002 | 88,9
29 | 1308831278 | 8,64E-003 | 115,7
30 | 951406034 | 6,28E-003 | 159,2
31 | 711322520 | 4,70E-003 | 212,9
32 | 552607034 | 3,65E-003 | 274,1
33 | 364782480 | 2,41E-003 | 415,2
34 | 260676436 | 1,72E-003 | 581,1
35 | 207373968 | 1,37E-003 | 730,4
36 | 131210384 | 8,66E-004 | 1.154,4
37 | 82503140 | 5,45E-004 | 1.836,0
38 | 68674194 | 4,53E-004 | 2.205,7
39 | 45318864 | 2,99E-004 | 3.342,4
40 | 27388352 | 1,81E-004 | 5.530,6
41 | 17060220 | 1,13E-004 | 8.878,7
42 | 9901676 | 6,54E-005 | 15.297,7
43 | 12150672 | 8,02E-005 | 12.466,2
44 | 4735376 | 3,13E-005 | 31.987,6
45 | 967920 | 6,39E-006 | 156.493,5
46 | 3685756 | 2,43E-005 | 41.096,9
47 | 1777896 | 1,17E-005 | 85.198,0
48 | 26048 | 1,72E-007 | 5.815.157,2
49 | 1208304 | 7,98E-006 | 125.360,2
50 | 0 | 0,00E+000 | ---------
51 | 69168 | 4,57E-007 | 2.189.932,0
52 | 0 | 0,00E+000 | ---------
53 | 0 | 0,00E+000 | ---------
54 | 260234 | 1,72E-006 | 582.065,4


note there is 2 3bv board every about 20,000 , so I guess everybody can get it if plays assidously. And there is 3 3bv board every 4,000, meaning you can get it in few months


PS: anybody knows how to make decent tables in this forum? I put ALL those vertical lines one by one! Tabulation does'nt work

[Tjips]

Ok, this might not be too big a deal, but I decided to play with this result a bit to get a usable equation describing this distribution. I did a Fourier transform on the data and constructed the an approximate Fourier series for this set. It looks like this:

f(u) = 0.0165 + (0.012cos(u) - 0.028sin(u)) + (-0.013cos(2u) - 0.011sin(2u)) + (-0.005cos(3u) + 0.005sin(3u)) + (0.002cos(4u) + 0.001sin(4u))

with u = (2pi/54)x - pi, 0 < x < 54 (meaning -pi < u < pi)

The components I included are those which are greater than 0.001 (by eye)

I've not gotten around to checking how good the reproduction is. I'll get around to that later...

EDIT: Checked it.... it's very accurate (also attached)

Using Wolfram|Alpha, it gives this normalized distribution:

Fourier series reproduction of the normalized beginner 3BV distribution

beg_dist2.jpg (7.01 KiB) Viewed 6481 times