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:
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...
Exact distribution of beginner 3BV
Exact distribution of beginner 3BV
- Attachments
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- results.zip
- (461 Bytes) Downloaded 650 times
Last edited by Tjips on Wed May 12, 2010 3:01 pm, edited 2 times in total.
The number of minesweeper boards:
Exp: 140055249834355336357264746443955277014822625680974475320364702381803619892657792049596418323789908370400 (1.4e104)
Int: 13115156192346373485000211099954895788134532256 (1.3e46) &
Beg: 18934455246 (1.9e10)
Exp: 140055249834355336357264746443955277014822625680974475320364702381803619892657792049596418323789908370400 (1.4e104)
Int: 13115156192346373485000211099954895788134532256 (1.3e46) &
Beg: 18934455246 (1.9e10)
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- Posts: 9
- Joined: Thu Aug 27, 2009 7:49 pm
- Location: Italy, Sicily
Re: Exact distribution of beginner 3BV
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
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
My times: 2.05+20.17+78.76=100.98
18th in Italy ranking
18th in Italy ranking
Re: Exact distribution of beginner 3BV
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:
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:
The components I included are those which are greater than 0.001 (by eye)Fourier wrote:
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)
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:
- Attachments
-
- Fourier series comparison with real data
- Fourier series comparison.jpg (35.96 KiB) Viewed 7421 times
The number of minesweeper boards:
Exp: 140055249834355336357264746443955277014822625680974475320364702381803619892657792049596418323789908370400 (1.4e104)
Int: 13115156192346373485000211099954895788134532256 (1.3e46) &
Beg: 18934455246 (1.9e10)
Exp: 140055249834355336357264746443955277014822625680974475320364702381803619892657792049596418323789908370400 (1.4e104)
Int: 13115156192346373485000211099954895788134532256 (1.3e46) &
Beg: 18934455246 (1.9e10)