From the very time that I became aware of Covers or Wheels with guarantees I asked myself is this really for real for the simple reason that those that espoused these absurdities referred to the condition to meet to get the guarantee in a way as if it was some trifle easily overcome! Nothing could be further from the truth as is apparent when referring to the table below! There are no Jackpot Lotteries for pretty much a Pool less than 25. **Getting 3 right in say a Pool of 20 is way different to getting 3 right from the full Pool of numbers used in the Lotto game whether it be 25, 45 or 49.**

For a Cover or Wheel with Guarantee that uses the full Pool of integers for that Lotto game as in C(49,6,3,6,1)=163 there is no **IF** - you are going to get a win every draw and damn the exorbitant cost for a pittance return! The syntax for this expression is - Cover(Max Integer, Pick, Prize Guarantee, Hits, Wins Guarantee)=Lowest Number of Combs. However, where the Pool is less than the maximum applicable to the game in this case 49 - **then some pretty big** **IFs** **come into play.**

Realize that the more constrictive the pool is the harder it is to get all the integers in that pool. For a 6/49 Lotto game where 6 integers are drawn the hardest is to get them in a Pool of 6 which is first prize followed by 7, 8 etc. This is why you need more combinations to get the guarantee as you increase the Pool size.

What's the point? Well, just like those lottery advertisements that say all you have to do to win is pick 6 numbers from say, 49 and then jump up and down for joy when the draw comes round so in Covers the big **IF** is often glossed over as if it is naturally going to happen. The reality is that those big **IFs** follow the odds and just don't happen that often.

Refer to the table below to get a reality check on just where the lower Pool sizes figure with the actual Lotto games' maximum Pool size. **In particular note that in a 6/49 game the combinations from Pool 44 are about 50% of the maximum so if you are dealing with say Pool 22 this represents about ½ of 1% NOT 45%.** **Easy on the caffeine and give up the sugar!**

In the table below pc6 (read as p choose 6) means all the combinations of 6 integers from the pool p. So, if p is 7 it is 7 combinations, if it is 8 then 28 combinations etc.

Pool(p) | pc6 -((p-1)c6) Combs | pc6Combs | pc6 - ((p-1)c6) Combs % of Pool 49 | Combs % of Pool 49 |

7 | 6 | 7 | 0.00004 | 0.00005 |

8 | 21 | 28 | 0.00015 | 0.00020 |

9 | 56 | 84 | 0.00040 | 0.00060 |

10 | 126 | 210 | 0.00090 | 0.00150 |

11 | 252 | 462 | 0.00180 | 0.00330 |

12 | 462 | 924 | 0.00330 | 0.00661 |

13 | 792 | 1,716 | 0.00566 | 0.01227 |

14 | 1,287 | 3,003 | 0.00920 | 0.02147 |

15 | 2,002 | 5,005 | 0.01432 | 0.03579 |

16 | 3,003 | 8,008 | 0.02147 | 0.05727 |

17 | 4,368 | 12,376 | 0.03124 | 0.08850 |

18 | 6,188 | 18,564 | 0.04425 | 0.13275 |

19 | 8,568 | 27,132 | 0.06127 | 0.19402 |

20 | 11,628 | 38,760 | 0.08315 | 0.27718 |

21 | 15,504 | 54,264 | 0.11087 | 0.38805 |

22 | 20,349 | 74,613 | 0.14552 | 0.53357 |

23 | 26,334 | 100,947 | 0.18832 | 0.72188 |

24 | 33,649 | 134,596 | 0.24063 | 0.96251 |

25 | 42,504 | 177,100 | 0.30395 | 1.26646 |

26 | 53,130 | 230,230 | 0.37994 | 1.64640 |

27 | 65,780 | 296,010 | 0.47040 | 2.11680 |

28 | 80,730 | 376,740 | 0.57731 | 2.69411 |

29 | 98,280 | 475,020 | 0.70281 | 3.39693 |

30 | 118,755 | 593,775 | 0.84923 | 4.24616 |

31 | 142,506 | 736,281 | 1.01908 | 5.26524 |

32 | 169,911 | 906,192 | 1.21505 | 6.48029 |

33 | 201,376 | 1,107,568 | 1.44006 | 7.92036 |

34 | 237,336 | 1,344,904 | 1.69722 | 9.61758 |

35 | 278,256 | 1,623,160 | 1.98984 | 11.60742 |

36 | 324,632 | 1,947,792 | 2.32148 | 13.92890 |

37 | 376,992 | 2,324,784 | 2.69592 | 16.62482 |

38 | 435,897 | 2,760,681 | 3.11715 | 19.74197 |

39 | 501,942 | 3,262,623 | 3.58945 | 23.33142 |

40 | 575,757 | 3,838,380 | 4.11731 | 27.44873 |

41 | 658,008 | 4,496,388 | 4.70550 | 32.15423 |

42 | 749,398 | 5,245,786 | 5.35904 | 37.51327 |

43 | 850,668 | 6,096,454 | 6.08323 | 43.59650 |

44 | 962,598 | 7,059,052 | 6.88366 | 50.48016 |

45 | 1,086,008 | 8,145,060 | 7.76618 | 58.24633 |

46 | 1,221,759 | 9,366,819 | 8.73695 | 66.98328 |

47 | 1,370,754 | 10,737,573 | 9.80243 | 76.78571 |

48 | 1,533,939 | 12,271,512 | 10.96939 | 87.75510 |

49 | 1,712,304 | 13,983,816 | 12.24490 | 100.00000 |

Regards

Colin Fairbrother

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Colin Fairbrother

These guaranteed Percentage Return figures (ie Benefit/Cost * 100) are based on current best optimized Covers for a 6/49 Lotto game with a cost per ticket or line set at 50¢ and with payouts set at $2.00 for a Three prize, $40.00 for a Four prize and $1,500 for a Five prize. Any variation with the prize payouts will not make much difference.

Pool | 3if6PR | 3if5PR | 3if4PR | 3if3PR | 4if6PR | 4if5PR | 4if4PR | 5if6PR |

49 | 2.45% | 0.89% | 0.74% | 7.33% | 2.03% | 1.04% | 0.43% | 2.02% |

The reason the 3if3 Percentage Return stands out is because you get mainly 20 threes per draw for an outlay of some $546.00 per draw which no one in their right mind would do.. For 1092 plays or tickets probability says 1 Four (which pays as much as 20 threes) and 19 threes - you can see the distortion ie You should be looking at 1 Four prize and 19 Three prizes to give about 14.5% but you don't have a guarantee on the Four - for that you need to outlay some $2,000.00 and then your return goes down to 2%. **Therein lies the reason why I think an obsession with finding the minimum set of numbers to guarantee the lowest tier prize as in a C(49,6,3,6,1) Cover works against you. The thrust should be on maximising the return as in any game of chance. I can think of no other game where the basic concept of Covers ie a win at any cost and at every participation - is promoted - but rather the concept of win some lose some but be ahead in the long term if possible is the norm. If you are a regular player of Lotto then unless you win first prize you are doomed pretty well to be on a losing streak - the best you can hope for is to mitigate your losses along the way.**

Without a doubt Lotto or Lotteries are one of the biggest cons perpetrated against the wagering public by Governments with payouts for the lower prizes that are some 30 times less than that obtainable from what are considered less attractive forms of gambling. They are monopolies and if subjected to competition from the private sector would not exist in their current form. Any self respecting Company charging the amount Government run Lotteries do for the return given would be held up to public scorn, derision and boycott. This con is only topped by the myriad of shysters, swindlers, slitherers, con artists, dirt bags and charlatans (I'm sure I left someone out) that misrepresent Covers or Wheels as returning ludicrously more than that which can be demonstratably proven.

Regards

Colin Fairbrother

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Colin Fairbrother

A Cover is only a Cover when the final combination is added to whatever lines it took to give the guarantee. Once you have a Cover that guarantees, say, that you will get a single three prize if you have 6 of the integers in a 6/49 Lotto ie C(49,6,3,6)=163 - then your reward is far short of what you can expect to get on average in lower prize wins according to probability - in fact about a third. Even if you spend a high amount per draw as with a Cover that guarantees you pretty well 20 threes per draw (C(49,6,3,3)=1222 this is still only in accordance with or less than the odds for threes for around that number of tickets.

To achieve the Cover in the minimum number of lines distortions are introduced which effect the yield. For example to get the 163 lines for a guarantee of a Three prize in the 6/49 Lotto game a Steiner system is used C(22,6,3,3)=77 which certainly gets as many Threes as possible into 77 lines (1540) but does not give the best coverage for all the 13,983,816 combinations of six integers for that number of lines. ie only around 8% compared to 88% with still 1540 unique Threes.

The construction of such a minimun line cover using whatever distortions means it can't be used progressively on a partial basis.**Randomize the 163 line 6/49 cover and the coverage fo 77 lines is only 56% a very poor result indeed.** Just as a broken bike chain is useless and when usable is only as strong as the weakest link, so a set of numbers that is distortingly produced is inferior to random selections where the prescribed intent and purpose is to maximize the yield**.**

The construction of such a minimun line cover using whatever distortions means it can't be used progressively on a partial basis.

Taking the first 77 Combinations of the 163 line Cover we have an inferior 56.19136% progress score towards full coverage if we get 6 integers matching the winning number. Compare this with generating 77 Combs 3 if 3 using the full 49 integers (with no duplicate Threes) which gives 64.07404% Coverage for the 3 if 6. If you go the extra distance and maximize the coverage while still maintaining no duplicate Threes as in my *Unique 3's* ™ then for the 77 Combs you get a whopping 86.88079% Coverage. To twist the knife, further progression shows an increasing disparity due to the insane obsession of those who wish to achieve a stupid objective and who are prepared to distort and reduce the percentage return no matter what the cost - say no more Colin.

Regards

Colin Fairbrother

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Colin Fairbrother

by

I have always been perplexed since coming across the term covers where it meant something other than what you slept under, as to what their real use is? As I see it the essence of a Cover is to absolutely **guarantee** a particular outcome from an event at the lowest expenditure. A search on the web throws up the term "Coverism" as being a method of prayer; I assume they cover themselves when praying. This is something akin to the proverbial ostrich burying its head in the sand when danger threatens and my overall assessment of Covers in relation to Lotto is that they are quite simply a con trick and ignore the reality of percentage return. In other words **there is no need for a guarantee of say the minimum prize every time you play Lotto** if the cost is some three times higher than what is delivered in due course by the odds.

I know Covers come from set theory in mathematics where they have respectability but what is their real use other than hoodwinking and bamboozling Lotto "aficionados" and providing a manna for swindlers, con-artists and just plain ignoramuses? Is it something to do with packing or networking or nodes, whatever?

Let's take the "famous" 163 lines for a 6/49 Lotto which guaranties 1 lousy three hit if you play it ie a $2 win even though it has cost you about $81.50. Now, let's do 10 runs of 40 draws with 4 random selection plays per draw ie 160 plays per run - so we're giving away 3 plays per run or 30 plays overall for the 10 runs. Here are the results warts and all ie no tinkering or discarding -

Run Three Win Four Win

1 4 -

2 2 -

3 2 -

4 - -

5 5 -

6 4 -

7 5 -

8 3 -

9 3 1

10 __ 1 __ __ - __

Totals 29 1

Looking at these results we see that only once out of 10 did we fare less than the 163 Line Cover by an insignificant 50¢. For 8 of the 10 runs we were ahead - fivefold twice, fourfold twice, threefold once, twofold twice and for one run we blitzed the Cover by the equivalent of **twenty three fold** by getting a four! For the total 1,630 plays we got 29 three wins whereas the Cover only guaranties 10!. The average for each run is 2.9 Three Wins which is in accordance with probability calculations as is the single Four win.

Jackpot or Powerball Lotto is about winning the top or higher tier prizes not the token or consolation ones. **Let me spell it out again - if some gullible person is playing 163 lines at 50¢ per line costing $81.50 in a 6/49 Lotto game as in a C(49,6,3,6) with a guarantee of a three win ie $2 and you are playing the minimum number of plays required to participate, say - 4 plays or $2.00, then for each game you are ahead by $77.50 if you lose!** Sounds to me more like winning! **Now, as far as the first prize is concerned when the odds are 13,983,816 to 1 playing 4 lines or 163 is an insignificant difference.** **ie 0.00002% compared to 0.00116%**

I await with eagerness for some bright soul to elucidate me on my source of bewilderment.

Regards

Colin Fairbrother

Colin Fairbrother

A handy reference table calculated just for you!

Combinations 1 to 9 Integers from Number Pool 1 to 56by Colin Fairbrother www.lottoposter.com | ||||||||

N | Comb2 | Comb3 | Comb4 | Comb5 | Comb6 | Comb7 | Comb8 | Comb9 |

2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

3 | 3 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |

4 | 6 | 4 | 1 | 0 | 0 | 0 | 0 | 0 |

5 | 10 | 10 | 5 | 1 | 0 | 0 | 0 | 0 |

6 | 15 | 20 | 15 | 6 | 1 | 0 | 0 | 0 |

7 | 21 | 35 | 35 | 21 | 7 | 1 | 0 | 0 |

8 | 28 | 56 | 70 | 56 | 28 | 8 | 1 | 0 |

9 | 36 | 84 | 126 | 126 | 84 | 36 | 9 | 1 |

10 | 45 | 120 | 210 | 252 | 210 | 120 | 45 | 10 |

11 | 55 | 165 | 330 | 462 | 462 | 330 | 165 | 55 |

12 | 66 | 220 | 495 | 792 | 924 | 792 | 495 | 220 |

13 | 78 | 286 | 715 | 1287 | 1716 | 1716 | 1287 | 715 |

14 | 91 | 364 | 1001 | 2002 | 3003 | 3432 | 3003 | 2002 |

15 | 105 | 455 | 1365 | 3003 | 5005 | 6435 | 6435 | 5005 |

16 | 120 | 560 | 1820 | 4368 | 8008 | 11440 | 12870 | 11440 |

17 | 136 | 680 | 2380 | 6188 | 12376 | 19448 | 24310 | 24310 |

18 | 153 | 816 | 3060 | 8568 | 18564 | 31824 | 43758 | 48620 |

19 | 171 | 969 | 3876 | 11628 | 27132 | 50388 | 75582 | 92378 |

20 | 190 | 1140 | 4845 | 15504 | 38760 | 77520 | 125970 | 167960 |

21 | 210 | 1330 | 5985 | 20349 | 54264 | 116280 | 203490 | 293930 |

22 | 231 | 1540 | 7315 | 26334 | 74613 | 170544 | 319770 | 497420 |

23 | 253 | 1771 | 8855 | 33649 | 100947 | 245157 | 490314 | 817190 |

24 | 276 | 2024 | 10626 | 42504 | 134596 | 346104 | 735471 | 1307504 |

25 | 300 | 2300 | 12650 | 53130 | 177100 | 480700 | 1081575 | 2042975 |

26 | 325 | 2600 | 14950 | 65780 | 230230 | 657800 | 1562275 | 3124550 |

27 | 351 | 2925 | 17550 | 80730 | 296010 | 888030 | 2220075 | 4686825 |

28 | 378 | 3276 | 20475 | 98280 | 376740 | 1184040 | 3108105 | 6906900 |

29 | 406 | 3654 | 23751 | 118755 | 475020 | 1560780 | 4292145 | 10015005 |

30 | 435 | 4060 | 27405 | 142506 | 593775 | 2035800 | 5852925 | 14307150 |

31 | 465 | 4495 | 31465 | 169911 | 736281 | 2629575 | 7888725 | 20160075 |

32 | 496 | 4960 | 35960 | 201376 | 906192 | 3365856 | 10518300 | 28048800 |

33 | 528 | 5456 | 40920 | 237336 | 1107568 | 4272048 | 13884156 | 38567100 |

34 | 561 | 5984 | 46376 | 278256 | 1344904 | 5379616 | 18156204 | 52451256 |

35 | 595 | 6545 | 52360 | 324632 | 1623160 | 6724520 | 23535820 | 70607460 |

36 | 630 | 7140 | 58905 | 376992 | 1947792 | 8347680 | 30260340 | 94143280 |

37 | 666 | 7770 | 66045 | 435897 | 2324784 | 10295472 | 38608020 | 124403620 |

38 | 703 | 8436 | 73815 | 501942 | 2760681 | 12620256 | 48903492 | 163011640 |

39 | 741 | 9139 | 82251 | 575757 | 3262623 | 15380937 | 61523748 | 211915132 |

40 | 780 | 9880 | 91390 | 658008 | 3838380 | 18643560 | 76904685 | 273438880 |

41 | 820 | 10660 | 101270 | 749398 | 4496388 | 22481940 | 95548245 | 350343565 |

42 | 861 | 11480 | 111930 | 850668 | 5245786 | 26978328 | 118030185 | 445891810 |

43 | 903 | 12341 | 123410 | 962598 | 6096454 | 32224114 | 145008513 | 563921995 |

44 | 946 | 13244 | 135751 | 1086008 | 7059052 | 38320568 | 177232627 | 708930508 |

45 | 990 | 14190 | 148995 | 1221759 | 8145060 | 45379620 | 215553195 | 886163135 |

46 | 1035 | 15180 | 163185 | 1370754 | 9366819 | 53524680 | 260932815 | 1101716330 |

47 | 1081 | 16215 | 178365 | 1533939 | 10737573 | 62891499 | 314457495 | 1362649145 |

48 | 1128 | 17296 | 194580 | 1712304 | 12271512 | 73629072 | 377348994 | 1677106640 |

49 | 1176 | 18424 | 211876 | 1906884 | 13983816 | 85900584 | 450978066 | 2054455634 |

50 | 1225 | 19600 | 230300 | 2118760 | 15890700 | 99884400 | 536878650 | 2505433700 |

51 | 1275 | 20825 | 249900 | 2349060 | 18009460 | 115775100 | 636763050 | 3042312350 |

52 | 1326 | 22100 | 270725 | 2598960 | 20358520 | 133784560 | 752538150 | 3679075400 |

53 | 1378 | 23426 | 292825 | 2869685 | 22957480 | 154143080 | 886322710 | 4431613550 |

54 | 1431 | 24804 | 316251 | 3162510 | 25827165 | 177100560 | 1040465790 | 5317936260 |

55 | 1485 | 26235 | 341055 | 3478761 | 28989675 | 202927725 | 1217566350 | 6358402050 |

56 | 1540 | 27720 | 367290 | 3819816 | 32468436 | 231917400 | 1420494075 | 7575968400 |

Regards

Colin Fairbrother

]]>
Colin Fairbrother

For those familiar with Covers as applicable to Lotto the free Covermaster program by John Rawson is no doubt familiar; if not it can be downloaded here.

You can easily build a C(49,6,3,6)=163 line cover by merging a C(22,6,3,3)=77 with a C(27,6,3,4)=86. This cover will guarantee a Three win every draw as at least one three in each of the 13,983,816 sixes is covered by the 163 line set.

The C(22,6,3,3)=77 is easy to build in CoverMaster - just don't check auto optimize - and has all the 1540 combinations of three that can be formed from a pool of 22 integers. This is a one off bonus and is why for the larger 3if6 covers these 77 lines are always used. In other words it is maximum packed for the number of lines with the possible distinct threes. As each six can only hold 20 threes (77x20=1540) it is impossible to do it in a lesser number of lines. You should note though that only 15 lines are needed for a C(22,6,3,6). Also note that for the 2600 combinations of 6 from a pool of 26 a C(26,6,3,3)=130 can be obtained (130x20=2600).

The C(27,6,3,4)=86 is impossible to the best of my knowledge and attempts to produce in CoverMaster without some manual intervention for the last 275 or so. There are lots of ways you can go about it eg Generate some 77 lines or so with your hits parameter set to 3 and then do a bit of manual scrutiny using whatever you have to assist - Access, Excel, MSQuery etc or you can do it in CoverMaster using techniques that don't seem to be well known in the Lotto crowd.

Using Covermaster optimization has only got me to 99.5% so far for a C(27,6,3,4)=86 so you are forced to use other means. You can try building it manually from scratch - just load your parameters - Pool, Pick, Match (Prize or Guarantee) then test and you will have a list of fours appear in a popup. Change the max. entries to a higher number than 25 and test again to get a bigger list of fours. From the menu bar Tools|Search|Best Play will give a list of sixes with coverage. Initially, the 435 means that in the list of uncovered fours using the 20 threes as criteria from that six a coverage of 435 is attainable. The difficulty lies in choosing which six - given that quite a few can have equal coverage. Which six you choose has repercussions on the later makeup of the overall set. After building the 163 line cover in Covermaster choose from the menu

Tools|Reports|Quick to get the following report.

Prize Matchif Hits in Pool | Tested | Covered | % Covered | Uncovered | % Uncovered |

2 If 2 | 1,176 | 582 | 49.489800% | 594 | 50.5102 |

2 If 3 | 18,424 | 18,424 | 100% | 0 | 0 |

2 If 4 | 211,876 | 211,876 | 100% | 0 | 0 |

2 If 5 | 1,906,884 | 1,906,884 | 100% | 0 | 0 |

2 If 6 | 13,983,816 | 13,983,816 | 100% | 0 | 0 |

2 If 7 | 85,900,584 | 85,900,584 | 100% | 0 | 0 |

3 If 3 | 18,424 | 3,007 | 16.321100% | 15,417 | 83.6789 |

3 If 4 | 211,876 | 98,719 | 46.592820% | 113,157 | 53.40718 |

3 If 5 | 1,906,884 | 1,570,086 | 82.337780% | 336,798 | 17.66222 |

3 If 6 | 13,983,816 | 13,983,816 | 100% | 0 | 0 |

3 If 7 | 85,900,584 | 85,900,584 | 100% | 0 | 0 |

4 If 4 | 211,876 | 2,428 | 1.145950% | 209,448 | 98.85405 |

4 If 5 | 1,906,884 | 102,539 | 5.377310% | 1,804,345 | 94.62269 |

4 If 6 | 13,983,816 | 2,016,925 | 14.423280% | 11,966,891 | 85.57672 |

4 If 7 | 85,900,584 | 22,898,862 | 26.657400% | 63,001,722 | 73.3426 |

5 If 5 | 1,906,884 | 978 | 0.051290% | 1,905,906 | 99.94871 |

5 If 6 | 13,983,816 | 42,157 | 0.301470% | 13,941,659 | 99.69853 |

5 If 7 | 85,900,584 | 884,838 | 1.030070% | 85,015,746 | 98.96993 |

6 If 7 | 85,900,584 | 7,009 | 0.008160% | 85,893,575 | 99.99184 |

Before considering this report in detail let's recap on the basic odds for a 6/49 Lotto game.

With one ticket your chances of getting the winning number are 1 in :

49!/(43!*6!) = 13,983,816

or 49c6

If you buy two tickets then bearing in mind there is only one winning ticket, dividing the 13,983,816 possible numbers in no particular order into two sets of numbers each containing 6,991,908 numbers you are guaranteed the winning number will be in one of the sets but it's still just a ratio. If you continued doing this ie divide the 13,983,816 combinations into 163 sets of around 85,790 then you know the winning six is in one of these sets but if you forked out some $42,900 to play one of these sets your chances of getting the winning number are only 1 in 163. Multiplying this by the extreme Monte Carlo factor of 26 (black came up 26 times in succession in 1913) you could spend some $182 million to get a million or so!

Your chances of getting a :

- five are 1 in 49c6/(6c5 * 43c1) = 54,201
- four are 1 in 49c6/(6c4 * 43c2) = 104
- three are 1 in 49c6/(6c3 * 43c3) = 56.6

The chances of getting a three of 1 in 56.6 in relation to the 163 line cover is very relevant. Put another way we are expecting 3 of the three prize wins on average every 170 plays. **Now, the 163 line cover may guarantee just 1 of these three prizes but good old Mr Random is relied upon for the other 2.** As I have pointed out before there is no appeal in paying some $82 to be guaranteed $2 with no increase in your chances of a higher prize - see Test C(49,6,3,6). **If you played a 56 set of numbers with a good spread of the threes then you will average a three win every draw and even if you played just a 28 set of numbers you would average a three win every other draw. **

Firstly, looking at the Covermaster report we see that Covermaster does validate the cover and this is a handy feature as one can use it to verify any Cover. If you delete lines and use Menu|Tools|Search|Best Play to reconstruct the cover the limitations of the algorithm used soon become apparent. Yes, with one or two or even 10 selections deleted it can reconstruct to 163 lines but try deleting the last 10 lines and you end up with a 166 line cover.

Secondly, from the 18,424 possible threes only 3,007 are included in the cover ie 16.32%. So, for every draw 83.68% of the threes are excluded ie if we divide our threes into roughly 6 bands then using the cover we are confining ourself to 1 band with a particular mathematical relationship for each random draw where all integers or threes are given an equal chance.

If you look at Menu|Tools|Reports|Detailed this will show you the repetition of threes in the cover. Each six can hold 20 threes. **Of the 3,007 threes used to make the cover:**

**2,828 appear once****142 twice****11 three times****18 four times****5 five times****3 six times**

A valid question to ask is if you are really after a few threes wouldn't it be better to do a random selection of 163 lines from my Unique Threes 450 line Cover which has 9000 distinct threes? **In other words you would have 163 lines with 3,260 distinct threes.**

The first line used in constructing any cover for a 6/49 Lotto covers 260, 624 sixes. ie at least 1 three in the six. For example using 01 02 03 04 05 06 we are covering all the 15,180 '01 02 03' threes and for the other 19 threes 3x14,190, 6x 13,244 and 10x12,341. Compare this with Menu|Tools|Reports|Charts|Dependency where we see that 5 of the sixes included are covering only between 3,000 to 4,000 sixes. Does the thought cross your mind that choosing a six made up from the 84% of the threes unused has some merit?

Covers may be used as an alternative to random selections when choosing one's lotto numbers but they do not perform over a reasonable period of time any better. There is a challenge in creating some of the larger covers and where there is a will there is a way for Covers come from the discrete side of mathematics - which means all the different ways it can be done are known and if tested then the optimum result can be obtained.

Below is the detailed report from CoverMaster: -

Win 6 Win 5 Win 4 Win 3 Comb 6's Comb 6's % Cum %

----------------------------------------------------------------------------

- - - 1 3595746 25.71362 25.71362

- - - 2 3261643 23.32441 49.03804

- - - 3 1114905 7.97282 57.01086

- - - 4 2858548 20.44183 77.45269

- - - 5 476971 3.41088 80.86357

- - - 6 223556 1.59868 82.46225

- - - 7 93980 0.67206 83.13431

- - - 8 64089 0.45831 83.59262

- - - 9 37278 0.26658 83.85920

- - - 10 220944 1.58000 85.43920

- - - 11 11270 0.08059 85.51979

- - - 12 3665 0.02621 85.54600

- - - 13 1290 0.00922 85.55522

- - - 14 348 0.00249 85.55771

- - - 15 139 0.00099 85.55871

- - - 16 29 0.00021 85.55891

- - - 17 6 0.00004 85.55896

- - - 18 8 0.00006 85.55901

- - - 20 2476 0.01771 85.57672

- - 1 0-17 1781368 12.73878 98.31550

- - 2 1-15 165076 1.18048 99.49598

- - 3 1-13 25917 0.18534 99.68132

- - 4 0-10 2138 0.01529 99.69661

- - 5 2-7 207 0.00148 99.69809

- - 6 0-7 53 0.00038 99.69847

- - 7 3-5 8 0.00006 99.69852

- - 9 2 1 0.00001 99.69853

- 1 0-4 0-15 41942 0.29993 99.99846

- 2 1-2 3-8 44 0.00031 99.99878

- 3 0 4-7 8 0.00006 99.99883

1 0 0-3 0-13__ 163__ 0.00117 100.00000

13,983,816

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- - - 1 3595746 25.71362 25.71362

- - - 2 3261643 23.32441 49.03804

- - - 3 1114905 7.97282 57.01086

- - - 4 2858548 20.44183 77.45269

- - - 5 476971 3.41088 80.86357

- - - 6 223556 1.59868 82.46225

- - - 7 93980 0.67206 83.13431

- - - 8 64089 0.45831 83.59262

- - - 9 37278 0.26658 83.85920

- - - 10 220944 1.58000 85.43920

- - - 11 11270 0.08059 85.51979

- - - 12 3665 0.02621 85.54600

- - - 13 1290 0.00922 85.55522

- - - 14 348 0.00249 85.55771

- - - 15 139 0.00099 85.55871

- - - 16 29 0.00021 85.55891

- - - 17 6 0.00004 85.55896

- - - 18 8 0.00006 85.55901

- - - 20 2476 0.01771 85.57672

- - 1 0-17 1781368 12.73878 98.31550

- - 2 1-15 165076 1.18048 99.49598

- - 3 1-13 25917 0.18534 99.68132

- - 4 0-10 2138 0.01529 99.69661

- - 5 2-7 207 0.00148 99.69809

- - 6 0-7 53 0.00038 99.69847

- - 7 3-5 8 0.00006 99.69852

- - 9 2 1 0.00001 99.69853

- 1 0-4 0-15 41942 0.29993 99.99846

- 2 1-2 3-8 44 0.00031 99.99878

- 3 0 4-7 8 0.00006 99.99883

1 0 0-3 0-13

13,983,816

**A Cover or Wheel with Guarantee is a set of numbers with each line having the same quantity of integers as the main integers picked in a draw such that it will give a nominated minimum number of prizes in each draw according to the condition that must be met of having so many correct in the pool of integers nominated. For C(49,6,3,6,1)=163 -**

.**The Pool is 49 integers. Each draw 6 main integers are picked.**

** **

A Three win is the minimum prize.

Six integers must be correct in the Pool which in this case is sufficient to form a winning number so all the 13,983,816 combinations that could be the winning Six must have a Three in the 163 line set.

For each draw at least 1 combination from the set must give at least the minimum prize of a Three. More prizes may be achieved but in every case the guarantee must be met.

There are 163 numbers, lines, blocks or plays in the set.

**A typical random selection of 163 lines would for about 720,000 of the possible 13,983,816 combinations of 6 integers not give a prize in each draw ie 5%. The C(49,6,3,6,1)=163 is about forcing all the combinations to give a prize but is it worthwhile?**

.

**From the probability calculation a randomly selected 163 line set would average 2.9 threes per draw in a 6/49 Lotto game. In fact I took the first 163 draws in my****All World's 6/49 History Database****and found for all the 13,983,816 combinations where Threes only were prizes that the average was 2.86. Comparing this with a 163 line Unique 3's™ number set and the 163 line Cover shows in both cases the Threes returned for 85% of the 13,963,616 combinations the same average of 2.7 per draw. However, for 85% of the possible combinations with Three only prizes the 163 Line Random Selection returned 2.85. In other words the guarantee makes no enhancement to the overall Three prizes and removes any reason for playing such a high number of combinations.**

.-
**A 163 Line set with maximum repeat of the 5's gives 11,142,652 Combinations where no prize is obtained 79.68% and for all the 3's only winnings this is done in 2,497,070 Combinations or 18%. The average then for the 3's hits with no other prizes is 1.22 well down on the other yields. Excluding the 1st prize there is 1 combination where you could walk away with 45 5's and 118 4's for about $73,220. Definitely not a recommended method of play.****.** **In a 6/49 Lotto game a Five prize should be worth around 957 times that paid for a Three prize and that for a Four prize around 18 times that paid for a Three prize. A check of the payouts from the Lottery operators will probably find $2.00 for a Three, $40.00 for a Four and a short changed $1500 or so for a Five. In other words getting a Four is worth 20 Threes and getting a Five is worth around 750 Threes. You can lose a few threes but never throw away your chances with the Fours and Fives by duplicating them in the number set you play.**

.**To guarantee the Three prize win every draw the 163 line cover has been structured such that apart from the 3's duplications (142x2;11x3;18x4;5x5;3x6) there is an unavoidable duplication of 17 of the 4's subsets. This is the important difference between a 163 line Unique 3's™ number set and the 163 line Cover. In other words as I have been saying since I first posted on the web it comes down to maximizing your chances with the Fives and Fours.**

Regards

Colin Fairbrother

__COVERS FOR POOL 7__

In considering these covers you should bear in mind they are extremely hypothetical as there is no Pick 6 Lotto game with just 7 integers - more like 45 or 49. So, to get 3 or more right from a 6/45 or 6/49 Lotto game in just a 7 integer pool that you nominate is a very rare occurrence. Some including me do say that to predicate something happening from just 7 Combinations when there are some 8 or 14 million other possible combinations is nothing less than farcical!

Here are the tables of Combinations that we need. In the covered column for doing this manually you would cross out the "No" when covered or if you were doing this programatically it would be changed to true.

**C(7,6,6,6) = 76 win if 6 of the integers from a pool of 7 are in the winning 6.**

This is a System 7 or Full Wheel which is all the combinations of 6 from a pool of 7 numbers as shown in table Sixes from Pool 7.

Now, if you manage to get 6 right not only do you get the winning six but also the other 6 combinations each have a five in them - so you do quite well on the night for an outlay of say, $3.50 or so.

If you only get 5 correct in the winning six you get 2 fives and 5 fours. If only 4 correct you get 3 fours and 4 threes. If only 3 correct you still get 4 threes.

**C(7,6,5,6) = 15 win if 6 of the integers from a pool of 7 are in the winning 6.**

Any one of the 7 Combinations of Six from the pool of 7 numbers will fulfill this cover by having at least one five in the other 6 combinations.

This cover highlights the drawbacks of covers that are other than System or Full Wheels. If on the night you see you have 6 integers correct from your pool of 7 integers then your jumping up and down and whooping should be restrained to that appropriate for a 5 win until you confirm that your one and only 6 is the winning six - then you can do cartwheels on the ceiling.

From 6 numbers we can obtain 6 combinations of 5. Let's say the winning 6 is:

01 02 03 04 05 06

and the combination we played was:

01 02 03 04 05 07

then of the 6 fives we can form from our play -

01 02 03 04 05

01 02 03 04 07

01 02 03 05 07

01 02 04 05 07

01 03 04 05 07

02 03 04 05 07

we see there is a match with 01 02 03 04 05 in the fives formed from the winning six -

Fives in our Six ID for Six Covered

01 02 03 04 05 1

01 02 03 04 06 3

01 02 03 05 06 4

01 02 04 05 06 5

01 03 04 05 06 6

02 03 04 05 06 7

(A simple way I use in forming these sets is to drop a number from right to left for each line.)

I deliberately started with ID 7 Six because any one of them has 15 fours and it's better to be flexible and experimental in ones thinking rather than regimented. You can learn by making mistakes. With the small pool of 7 numbers we don't have a lot of choices and I simply formed the sixes progressively as required.

You're forgiven if you are starting to think - is that all it is! All those Guru Guru people poncing about and making out they're ever so clever and a 9 year old can understand it. (Is that right Sash? Yes, Dad! There I told you!)

**C(7,6,3,4) = 13 win if 4 of the integers from a pool of 7 are in the winning 6.**

Each Six contains 15 fours. Check out for yourself and you will see any of our 7 sixes from a pool of 7 integers will cover the 35 distinct fours that can be formed from the same pool. eg The Six in which the Four 01 02 03 07 appears is covered by 01 02 03 04 05 06 because the other two integers in that six must be 04 05 or 04 06 or 04 07 or 05 06 or 05 07 or 06 07 in any case providing the other required integer.

**C(7,6,3,3) = 43 win if 3 of the integers from a pool of 7 are in the winning 6.**

Regards

Colin

ps They do get harder.

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My approach to Covers or Wheels with a prize guarantee and in fact most of what I do in relation to Lotto is that of a programmer with relational database skills - we have our own bag of tricks and ways of approaching tasks which can't easily be simulated with arrays without much tedious work and advanced coding. This goes way beyond Microsoft Excel which is most commonly used in this field of Lotto number analysis with its "flat file" constraints.

Regularly, I see on other Forums enthusiasts struggling to get Excel to do something which is relatively easy to accomplish once the GRID is dropped - grids are great for spreadsheets and jails, but not good for creative programming. Like most things there is an initial steep learning curve - but once you get over that, well it doesn't necessarily get easier because so many new vistas open up that you're on a never ending learning curve.

Over thirty years of troubleshooting has taught me never to forget the **BIG PICTURE** - don't get bogged down in detail, know enough to know where to look for the detail, look for the simplest solution first and never be timid about using wizards or tools to get the job done quicker and easier ie more efficiently.

Covers for a small pool of numbers can be done manually - as you head into the bigger pools you need a computer tool such as CoverMaster or as in my case and other specialists in this field with programming knowledge we make our own and this requires advanced coding skills.

You will find an expression used with Coverings which is worth becoming familiar with and can easily be remembered by using the mnemonic PoPiMaHi ie Pool, Pick, Match and Hits. C(7,6,3,3)=4 simply means we are restricting our combinations to those formed from a pool of 7 integers in a Pick 6 Lotto game and we want aYou should bear in mind that where the pool is less than that used in the Lotto game then it is all hypothetical, some like me would say pure baloney as all the possible combinations of 6 from a pool of 7 is only 7 or for Pool 22 only 74,613 whereas a pool of 45 gives 8,145,060 and 49 gives 13,983,816 - in other words it is like a drop in the ocean or basing something on the assumption that the cow has already jumped over the moon. You will hear this garbage trotted out ad nauseam by those whose very ignorance is given away usually before they have completed one garbled sentence - but in their own peanut brains they live in a delusionary world where they think they are "gurus'.

01 02 03 04 05 06

01 02 03 04 05 07

01 02 03 05 06 07

01 02 04 05 06 07

01 03 04 05 06 07

02 03 04 05 06 07

However, this guarantee is meaningless when the same 7 lines are applied to a real Jackpot type Lotto game where the smallest Pool is 25 as in the West Virginia game. The token prize for getting 3 correct in the winning Comb6 is better achieved by randomizing the following template:

01 02 03 04 05 06

07 08 09 10 11 12

13 14 15 16 17 18

19 20 21 22 23 24

01 02 07 13 19 25

03 04 08 14 20 25

05 06 09 15 21 25

The difference is phenominal with the 7 line guarantee only covering 34,300 of the possible 177,100 combinations of 6 integers for the Pool 25 game or 19.37% for a match 3 integers whereas the latter tailor made 7 lines covers 122,671 or 69.27%. You may notice the following repetition in the guarantee lines: -

7 Comb1's repeated 6x

21 Comb2's repeated 5x

35 Comb3's repeated 4x

35 Comb4's repeated 3x

21 Comb5's repeated 2x

Contrast this with the 7 maximized coverage lines for Pool 25 where the paying Comb3's, Comb4's and Comb5's are not repeated at all.

We will look at the merits of partial or % Covers or Wheels, because you will see an obsession with meeting the full requirements of a Cover means you may be including a Comb 6 which has 20 threes in it just because 1 three is not covered. Contrast this with in a 6/49 Lotto at the outset of building up your cover any six will cover 260,624 sixes from the 13,983,816 - a significant difference.

The question is raised as to whether your percentage progress towards achieving the goal of a cover gives any merit to a partial set given that a high premium is paid for guaranteeing a partial entitlement in each game ie in a 6/49 Lotto the odds say you should get an average of 3 three prize wins every 170 plays whereas your guarantee is only 1 fom a 163 line C(49,6,3,6) cover.

Colin Fairbrother

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