Covering Pairs in 6/49
Judy
My aim is to cover all 1176 pairs in as few sets of six random numbers as
possible.
As each set of 6 numbers comprises 15 pairs I should be able to do this in a
theoretical 78.4 sets .
Being only a theoretical value I multiplied it by 4 and set up a program to
generate 300 sets at a time.
My problem is that , even after +/- 30k cycles, the best cover I can achieve
is +/- 1000 of the 1176 pairs i.e. even after generating 300*15 = 4500 pairs
each cycle.
I'm no statistician and would like to know if it is possible to calculate
the likely number of cycles needed before the magical 1176 is arrived at.
Any help appreciated
Mike
Frank
> I've been using XL with a [I think] pretty good RNG macro.
> My aim is to cover all 1176 pairs in as few sets of six random numbers as
> possible.
82 lines is the least amount of lines that allows you to get your
purposes:
49,6,2,2=82
1 2 3 4 5 6
1 2 3 7 8 9
1 10 11 12 13 14
1 15 16 17 18 19
1 20 21 22 23 24
1 25 26 27 28 29
1 30 31 32 33 34
1 35 36 37 38 39
1 40 41 42 43 44
1 45 46 47 48 49
2 10 25 30 35 45
2 11 15 31 36 46
2 12 20 32 37 47
2 13 33 38 40 48
2 14 16 21 26 41
2 17 22 27 42 49
2 18 23 28 34 43
2 19 24 29 39 44
3 10 16 24 34 46
3 11 19 21 40 45
3 12 18 22 39 48
3 13 17 23 25 47
3 14 27 32 38 43
3 15 28 33 37 42
3 20 26 31 35 44
3 29 30 36 41 49
4 5 6 7 8 11
4 9 17 26 34 37
4 10 18 20 29 38
4 12 16 27 36 40
4 13 19 28 35 49
4 14 23 33 44 45
4 15 24 30 43 48
4 21 31 39 42 47
4 22 25 32 41 46
5 9 23 27 30 39
5 10 22 28 31 40
5 12 24 26 33 49
5 13 15 21 29 32
5 14 18 35 42 46
5 16 25 37 44 48
5 17 20 36 43 45
5 19 34 38 41 47
6 9 18 32 44 49
6 10 17 33 39 41
6 12 19 25 31 43
6 13 16 20 30 42
6 14 24 28 36 47
6 15 22 26 38 45
6 21 27 34 35 48
6 23 29 37 40 46
7 9 20 28 41 48
7 10 15 27 44 47
7 12 29 34 42 45
7 13 26 39 43 46
7 14 19 22 30 37
7 16 23 31 38 49
7 17 24 32 35 40
7 18 21 25 33 36
8 10 21 37 43 49
8 11 24 25 38 42
8 12 15 23 35 41
8 13 22 34 36 44
8 14 17 29 31 48
8 16 28 32 39 45
8 18 26 30 40 47
8 19 20 27 33 46
9 10 19 36 42 48
9 11 16 22 35 43
9 11 16 29 33 47
9 12 21 28 38 46
9 13 24 31 41 45
9 14 15 20 25 40
10 19 23 26 32 36
11 17 28 30 38 44
11 18 24 27 37 41
11 20 25 34 39 49
11 23 26 32 42 48
12 17 21 30 44 46
13 18 27 31 37 45
14 15 34 39 40 49
22 29 33 35 43 47
> As each set of 6 numbers comprises 15 pairs I should be able to do this in a
> theoretical 78.4 sets .
> Being only a theoretical value I multiplied it by 4 and set up a program to
> generate 300 sets at a time.
> My problem is that , even after +/- 30k cycles, the best cover I can achieve
> is +/- 1000 of the 1176 pairs i.e. even after generating 300*15 = 4500 pairs
> each cycle.
> I'm no statistician and would like to know if it is possible to calculate
> the likely number of cycles needed before the magical 1176 is arrived at.
>
> Any help appreciated
>
> Mike
Look at this wheel as pointer numbers, order randomly the 49 numbers
and then substitute the pointer by the order your program gets. This
way you can have a lot of different blocks of 82 lines, every block
having the 1176 pairs inside.
Frank
lotbook
> purposes:
This wheel has been constructed (and its minimality proven) by
Bluskov-Greig-Heinrich in an article published in Can Math Bulletin
43(2000).
More Bluskov's minimal wheels can be found in his book
Combinatorial lottery systems; see
http://www3.telus.net/lotbook/