Combination problem

82
My "algorithm" is brute force - totally independent of IMAX. I gave up on a looping "step 3" that was limited to 36×36, but instead looked at the whole 45×45 at once, using a scheme to try to find maximum number of new pairs for each ticket. I started by accepting 12 or more, then when those dried up accepted tickets with fewer new pairs.

After looping by 1's and by 7's to get the first 15 tickets with 15 pairs each, the procedure is:
Code:
1. Find a value M that has been used in fewest pairs so far.
2. Find a value I which has not yet been paired with M.
3. Look for J that has not been paired with M or with I: 
   if none found, try another I with same M
   if no more (M,I) pairs can be found, try a different M
4. For all K, count number of new pairs in set (M,I,J,K) and use K that is max
5. For all L, count number of new pairs in set (M,I,J,K,L) and use L that is max
6. for all N, count number of new pairs in set (M,I,J,K,L,N) and use N that is max
7. If number of new pairs [U]>[/U] current criterion, keep the ticket
   else try again from step 1.
8. If 10 in a row fail, decrease criterion by 1.
This method runs runs into trouble (that is, infinite loop) when the remaining number of pairs is less than 60 or so, so at that point I switch to finding pairs that haven't yet been used, and trying to find some with common values. About the last 10 tickets are found that way.

Every possible pair occurs in some ticket with 4 other "random" values. That makes the probability of having any specific set of 3 values P(3) = 4/43. Is that enough to skew the odds in your favor? Is there a payout for getting 3 hits?

So essentially, there is no true algorithm, but it's more a trial-and-error method?
There's no payout with getting 3-hits unfortunately. I've been playing this method for a few weeks now, more out of interest than to expect any significant return. I've actually won a smaller prize every week with my 94 games. Albeit, the system is not as simple as the math problem I proposed as there is actually 8 numbers drawn with 2 of those being considered as supplementary. I wasn't going to complicate things, as I had already withheld the second part of my question which was whether or not an algorithm could be found where, instead of pairs, groups of n>2 could also be found (ie: How many subsets required to cover all quadruples using subsets of 9, from a field of 200). This had nothing to do with lottery anymore, but was more just out of interest for me.
 
After playing around with a few more values, it would seem that repetition is unavoidable only when a:b is such that b^-0.5 = a.
ie: To do 6:45, one may be better off approaching it as a 6:36, then somehow working in the other 9.
Assuming that 6:36 can be done without repetition, this would represent 630 pairs from 42 games.
Then by grouping the numbers into triples, A=[37 38 39], B=[40 41 42], C=[43 44 45], as well as grouping the remaining 1 - 36 into triples (named D to O), we can play two sets of triples per subset to get 9 unique pairs, and 6 repetitions, for the remaining games.

The 3 triples (A, B, and C) are matched up with each of the other 12 triples (C through to O) to give 36 more games.
The 3 triples (A, B, and C) are then matched with each other for an additional 3 games.

This means, that if Denis' theory is right about no repetitions possible for a 'nice lotto', then 6:45 can be achieved in 81 games.

Seems I've made an error here.

990 combinations (45C6) - 630 combinations (from 42 subsets) = 360 remaining combinations.

360 / 9 (assuming this is the maximum number of unique combinations achievable per subset after the first 42 games) is 40.

42 + 40 = 82.

Maybe 82 is the minimum after all?
 
Maybe 82 is the minimum after all?
Here then is my set of 82
Code:
   N              TICKET             New  Sum
   1       1   2   3   4   5   6      15   15
   2       7   8   9  10  11  12      15   30
   3      13  14  15  16  17  18      15   45
   4      19  20  21  22  23  24      15   60
   5      25  26  27  28  29  30      15   75
   6      31  32  33  34  35  36      15   90
   7      37  38  39  40  41  42      15  105
   8      43  44  45   7  14  21      15  120
   9      28  35  42   4  11  18      15  135
  10      25  32  39   1   8  15      15  150
  11      22  29  36  43   5  12      15  165
  12      19  26  33  40   2   9      15  180
  13      16  23  30  37  44   6      15  195
  14      13  20  27  34  41   3      15  210
  15      10  17  24  31  38  45      15  225
  16       1   7  13  19  28  31      15  240
  17       2   7  15  20  29  35      15  255
  18       3   7  16  22  25  33      15  270
  19       4   7  17  23  26  32      15  285
  20       5   7  18  24  27  37      15  300
  21       6   7  34  38   8  14      13  313
  22       9   1  14  20  30  36      15  328
  23      10   1  16  21  26  34      15  343
  24      11   1  17  22  27  40      15  358
  25      12   1  18  23  33  38      15  373
  26      39   2  10  13  22  30      15  388
  27      41   1  24  29  44   4      14  402
  28      42   1  43   2   8  16      12  414
  29      45   1  35  37   3   8      13  427
  30       5   8  13  21  33   4      14  441
  31       6   9  13  24  25  35      15  456
  32      11   2  14  23  25  31      15  471
  33      12   2  17  21  28  37      15  486
  34      15   3   9  21  31  42      15  501
  35      19   3  10  14  29  32      15  516
  36      36   2  18  41  45   6      14  530
  37      39   3  11  24  26  36      15  545
  38      40   3  12  30   4  15      13  558
  39      43   3  17   6  19  39      12  570
  40      44   2  27  32  38   9      14  584
  41       5   9  16  28  39  45      15  599
  42      20   4  10  25  37  43      15  614
  43      40   5  10  23  35  41      14  628
  44      42   5  14  26  12  20      13  641
  45      15   5  11  19  34  44      15  656
  46      40   6  20  28  32   8      13  669
  47      18   3  44   8  22  26      12  681
  48      27   4  14  39  31  12      12  693
  49      29   6  11  21  33  27      12  705
  50      30   5  17   8  31  41      12  717
  51      36   4  16  19  38  25      13  730
  52      43   9  18  34  23  29      13  743
  53      45   4  22  34  42  25      12  755
  54      32  11  13  37  43  26      12  767
  55      33  10  15  28  24  43      13  780
  56      40   7  36  13  42  44      12  792
  57      31   6  22  15  26  37      11  803
  58      30  11  38  20  45  33      11  814
  59      35  12  16  19  27  41      11  825
  60      18  19  30  21  32  42      11  836
  61      35  14  22  28  38  41      11  847
  62      36   8  23  27  15  45      12  859
  63      40  14  24  16  29  31      10  869
  64      39  18  20  31  44  10      10  879
  65      32  12  24  34   2  30       9  888
  66      17  25  44  12  33  35       9  897
  67      41  21  25  40  18  43       9  906
  68      29  13  38  45  12  26       9  915
  69      39  21  35   7  30  43       8  923
  70      17  29  42   6  10  27       8  931
  71      28  34  37  39  40  36       6  937
  72       3  23  13  28  39  42       6  943
  73      11  16  20  32  41  15       6  949
  74      14  33  37  39  41  42       6  955
  75      19  37  29  45  32  40       5  960
  76       3  38   5  15  21  43       5  965
  77       4   9  17  22  37  41       5  970
  78       8  19  24  42  29  39       5  975
  79      26  35  41  27  43  31       4  979
  80       5  25  32  22   6  12       4  983
  81       7  41  10  36  17  21       4  987
  82      17  20  34  28  44  19       3  990
 
Distribution of # of new pairs:
   15   34
   14    5
   13    8
   12   10
   11    5
   10    2
    9    4
    8    2
    7    0
    6    4
    5    4
    4    3
    3    1
    2    0
    1    0
 
    3 pairs repeated 4 times
   21 pairs repeated 3 times
  189 pairs repeated 2 times
  777 pairs not repeated
    0 missing pairs
 
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