Colin Fairbrother

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whether other than through sheer luck the odds can be bettered.

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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Click article to view this still very informative piece by Uenal Mutlu even though a bit outdated because of lower bounds subsequently achieved.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
Lines

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
Lines

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


 


This article first published on the web 20/1/2010 and has been viewed 10220 times.
The author may be contacted at fairbros@bigpond.com. To generate sets of Lotto numbers to play for most Lotto games around the world go to LottoToWin ® where for a US$5.00 annual subscription you can store the numbers as well, to check later for wins. The numbers produced have no duplicate subset combinations of two integers (if applicable) or three integers, use all the integers for the particular Lotto game and maximize the coverage of the potential winning main number. The articles on this site have been selected from the many at the forum LottoPoster
© Colin Fairbrother 2004 to 2012 All rights reserved.