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| Reference Tables for Cover Upper Bounds & Merges | |
 LottoPoster Forums : WORLD RECORD LOTTO COVERS (WHEELS WITH GUARANTEE) : Reference Tables for Cover Upper Bounds & Merges |
 Topic: Merged Covers for Pick 6, Pool 49 Classic Lotto |
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Colin F
Lotto Systems Tester Creator & Analyst 
To dream the impossible dream ... Joined: September 30 2004 Location: Australia Online Status: Offline Posts: 678 |
   Topic: Merged Covers for Pick 6, Pool 49 Classic LottoPosted: January 20 2010 at 10:03pm |
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The seminal article on merged Covers to the writer from around 2000 when I became interested in this subject is that of Uenal Mutlu written in 1995 which introduced me to the maximum packing Steiner system C(22,6,3,3,1)=77 and when producing my own table for the merge possibilities to the C(26,3,3,1)=130 Steiner system.The merging of these two systems gives a C(48,6,3,5)=207 which is also a cover for C(49,6,3,5)=207 and of course also a C(49,6,3,6)=207. Uenal Mutlu's article is mainly about the merging of the Steiner C(22,6,3,3,1)=77 with at the time a C(27,6,3,4,1)=91 to give a C(49,6,3,6,1)=168. Steiner systems can't be reduced any more but the C(27,6,3,4,1) has been reduced to 86 so it is now done in 163 blocks.. . Click article to view this still very informative piece by Uenal Mutlu even though a bit outdated because of lower bounds subsequently achieved. Here is an extract from the article: -. "Theorem 3.1 includes also the well known two cases ... :
C(v1,6,3,3,1,=b1) + C(v2,6,3,3,1,=b2) = C(v1+v2,6,3,5,1,=b1+b2 C(v1,6,3,3,1,=b1) + C(v2,6,3,4,1,=b2) = C(v1+v2,6,3,6,1,=b1+b2" The writer holds the view which is at odds with the general view on Covers that the prize return in playing Lotto should be the main consideration and not distortion of the prize distribution just to guarantee what well may be an insignificant prize in relation to the amount outlayed. eg If playing 300 lines and 300 combinations are uncovered then the chances of the winning number being a non prize is the same as that of winning first prize. While I don't recommend playing the Covers in the tables below because of possibly the wide distribution range of the integers, the repetition of paying subset Threes and the usual distortion of the prize grouping towards the lowest prize or a non smooth falling away from the dominant Threes group (ie not a Bell Curve) they are a good indication of what is nearly achievable with better yielding Lotto Designs. A quick reference for the lower bounds likely to be of use in Pick 6 Lotto games is available here. Two good sources for mathematical Covers are Weef's site. or La Jolla.
3if6 Covers formed from merging two Covers for Pool 49, Pick 6 Lotto 3 if 3 3 if 4 Combined Pool Lines Pool Lines 8 4 41 321 325 9 7 40 301 308 10 10 39 276 286 11 11 38 261 272 12 15 37 249 264 13 21 36 229 250 14 25 35 205 230 15 31 34 189 220 16 38 33 172 210 17 44 32 149 193 18 48 31 133 181 19 63 30 119 182 20 72 29 108 180 21 77 28 97 169 22 77 27 86 163 23 104 26 76 180 24 116 25 58 174 25 130 24 61 191 26 130 23 54 184 27 167 22 46 213 28 188 21 40 228 29 221 20 35 256 30 225 19 30 255 31 273 18 25 298 32 300 17 20 320 33 334 16 16 350 34 364 15 14 378 35 407 14 11 418 36 420 13 9 429 37 479 12 6 485 38 513 11 5 518 39 539 10 4 543 40 576 9 3 579
3if5 Covers formed from merging two Covers for Pool 49, Pick 6 Lotto 3 if 3 3 if 3 Combined Pool Lines Pool Lines 8 4 41 634 638 9 7 40 576 604 10 10 39 539 549 11 11 38 513 524 12 15 37 479 494 13 21 36 420 441 14 25 35 407 432 15 31 34 364 395 16 38 33 334 372 17 44 32 300 344 18 48 31 273 321 19 63 30 225 288 20 72 29 221 293 21 77 28 188 265 22 77 27 167 244 22 77 26 130 207* 23 104 26 130 234 24 116 25 130 246 25 130 24 116 246 26 130 23 104 234 27 167 22 77 244 28 188 21 77 265 * Special case merging two Steiner Systems Pool 48 good for Pool 49.
Colin Fairbrother |
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