Covers or Wheels with Guarantees as applied to Lotto have concentrated on a single lowest value prize guarantee but are they giving you the best return? Follow this link multiple prize wins in Lotto for an article on the best way of achieving multiple prizes with particular attention to the remarkable attributes for pool 22.

a1 | a2 | a3 | a4 | a5 | a6 | t1 | t2 | t3 | t4 | t5 | t6 | t7 | t8 | t9 | t10 | t11 | t12 | t13 | t14 | t15 | t16 | t17 | t18 | t19 | t20 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

a | b | c | d | e | f | abc | abd | abe | abf | acd | ace | acf | ade | adf | aef | bcd | bce | bcf | bde | bdf | bef | cde | cdf | cef | def |

a | b | g | h | i | j | abg | abh | abi | abj | agh | agi | agj | ahi | ahj | aij | bgh | bgi | bgj | bhi | bhj | bij | ghi | ghj | gij | hij |

c | d | g | h | k | l | cdg | cdh | cdk | cdl | cgh | cgk | cgl | chk | chl | ckl | dgh | dgk | dgl | dhk | dhl | dkl | ghk | ghl | gkl | hkl |

e | f | i | j | k | l | efi | efj | efk | efl | eij | eik | eil | ejk | ejl | ekl | fij | fik | fil | fjk | fjl | fkl | ijk | ijl | ikl | jkl |

e | f | g | h | m | n | efg | efh | efm | efn | egh | egm | egn | ehm | ehn | emn | fgh | fgm | fgn | fhm | fhn | fmn | ghm | ghn | gmn | hmn |

c | d | i | j | m | n | cdi | cdj | cdm | cdn | cij | cim | cin | cjm | cjn | cmn | dij | dim | din | djm | djn | dmn | ijm | ijn | imn | jmn |

a | b | k | l | m | n | abk | abl | abm | abn | akl | akm | akn | alm | aln | amn | bkl | bkm | bkn | blm | bln | bmn | klm | kln | kmn | lmn |

c | e | g | i | o | p | ceg | cei | ceo | cep | cgi | cgo | cgp | cio | cip | cop | egi | ego | egp | eio | eip | eop | gio | gip | gop | iop |

d | f | h | j | o | p | dfh | dfj | dfo | dfp | dhj | dho | dhp | djo | djp | dop | fhj | fho | fhp | fjo | fjp | fop | hjo | hjp | hop | jop |

a | f | g | k | o | q | afg | afk | afo | afq | agk | ago | agq | ako | akq | aoq | fgk | fgo | fgq | fko | fkq | foq | gko | gkq | goq | koq |

b | e | h | l | o | q | beh | bel | beo | beq | bhl | bho | bhq | blo | blq | boq | ehl | eho | ehq | elo | elq | eoq | hlo | hlq | hoq | loq |

b | d | i | k | p | q | bdi | bdk | bdp | bdq | bik | bip | biq | bkp | bkq | bpq | dik | dip | diq | dkp | dkq | dpq | ikp | ikq | ipq | kpq |

a | c | j | l | p | q | acj | acl | acp | acq | ajl | ajp | ajq | alp | alq | apq | cjl | cjp | cjq | clp | clq | cpq | jlp | jlq | jpq | lpq |

b | c | j | k | o | r | bcj | bck | bco | bcr | bjk | bjo | bjr | bko | bkr | bor | cjk | cjo | cjr | cko | ckr | cor | jko | jkr | jor | kor |

a | d | i | l | o | r | adi | adl | ado | adr | ail | aio | air | alo | alr | aor | dil | dio | dir | dlo | dlr | dor | ilo | ilr | ior | lor |

a | e | h | k | p | r | aeh | aek | aep | aer | ahk | ahp | ahr | akp | akr | apr | ehk | ehp | ehr | ekp | ekr | epr | hkp | hkr | hpr | kpr |

b | f | g | l | p | r | bfg | bfl | bfp | bfr | bgl | bgp | bgr | blp | blr | bpr | fgl | fgp | fgr | flp | flr | fpr | glp | glr | gpr | lpr |

c | f | h | i | q | r | cfh | cfi | cfq | cfr | chi | chq | chr | ciq | cir | cqr | fhi | fhq | fhr | fiq | fir | fqr | hiq | hir | hqr | iqr |

d | e | g | j | q | r | deg | dej | deq | der | dgj | dgq | dgr | djq | djr | dqr | egj | egq | egr | ejq | ejr | eqr | gjq | gjr | gqr | jqr |

m | n | o | p | q | r | mno | mnp | mnq | mnr | mop | moq | mor | mpq | mpr | mqr | nop | noq | nor | npq | npr | nqr | opq | opr | oqr | pqr |

b | d | g | m | o | s | bdg | bdm | bdo | bds | bgm | bgo | bgs | bmo | bms | bos | dgm | dgo | dgs | dmo | dms | dos | gmo | gms | gos | mos |

a | c | h | n | o | s | ach | acn | aco | acs | ahn | aho | ahs | ano | ans | aos | chn | cho | chs | cno | cns | cos | hno | hns | hos | nos |

a | f | i | m | p | s | afi | afm | afp | afs | aim | aip | ais | amp | ams | aps | fim | fip | fis | fmp | fms | fps | imp | ims | ips | mps |

b | e | j | n | p | s | bej | ben | bep | bes | bjn | bjp | bjs | bnp | bns | bps | ejn | ejp | ejs | enp | ens | eps | jnp | jns | jps | nps |

c | e | k | m | q | s | cek | cem | ceq | ces | ckm | ckq | cks | cmq | cms | cqs | ekm | ekq | eks | emq | ems | eqs | kmq | kms | kqs | mqs |

d | f | l | n | q | s | dfl | dfn | dfq | dfs | dln | dlq | dls | dnq | dns | dqs | fln | flq | fls | fnq | fns | fqs | lnq | lns | lqs | nqs |

h | j | l | m | r | s | hjl | hjm | hjr | hjs | hlm | hlr | hls | hmr | hms | hrs | jlm | jlr | jls | jmr | jms | jrs | lmr | lms | lrs | mrs |

g | i | k | n | r | s | gik | gin | gir | gis | gkn | gkr | gks | gnr | gns | grs | ikn | ikr | iks | inr | ins | irs | knr | kns | krs | nrs |

a | e | j | m | o | t | aej | aem | aeo | aet | ajm | ajo | ajt | amo | amt | aot | ejm | ejo | ejt | emo | emt | eot | jmo | jmt | jot | mot |

b | f | i | n | o | t | bfi | bfn | bfo | bft | bin | bio | bit | bno | bnt | bot | fin | fio | fit | fno | fnt | fot | ino | int | iot | not |

b | c | h | m | p | t | bch | bcm | bcp | bct | bhm | bhp | bht | bmp | bmt | bpt | chm | chp | cht | cmp | cmt | cpt | hmp | hmt | hpt | mpt |

a | d | g | n | p | t | adg | adn | adp | adt | agn | agp | agt | anp | ant | apt | dgn | dgp | dgt | dnp | dnt | dpt | gnp | gnt | gpt | npt |

g | i | l | m | q | t | gil | gim | giq | git | glm | glq | glt | gmq | gmt | gqt | ilm | ilq | ilt | imq | imt | iqt | lmq | lmt | lqt | mqt |

h | j | k | n | q | t | hjk | hjn | hjq | hjt | hkn | hkq | hkt | hnq | hnt | hqt | jkn | jkq | jkt | jnq | jnt | jqt | knq | knt | kqt | nqt |

d | f | k | m | r | t | dfk | dfm | dfr | dft | dkm | dkr | dkt | dmr | dmt | drt | fkm | fkr | fkt | fmr | fmt | frt | kmr | kmt | krt | mrt |

c | e | l | n | r | t | cel | cen | cer | cet | cln | clr | clt | cnr | cnt | crt | eln | elr | elt | enr | ent | ert | lnr | lnt | lrt | nrt |

d | e | h | i | s | t | deh | dei | des | det | dhi | dhs | dht | dis | dit | dst | ehi | ehs | eht | eis | eit | est | his | hit | hst | ist |

c | f | g | j | s | t | cfg | cfj | cfs | cft | cgj | cgs | cgt | cjs | cjt | cst | fgj | fgs | fgt | fjs | fjt | fst | gjs | gjt | gst | jst |

k | l | o | p | s | t | klo | klp | kls | klt | kop | kos | kot | kps | kpt | kst | lop | los | lot | lps | lpt | lst | ops | opt | ost | pst |

a | b | q | r | s | t | abq | abr | abs | abt | aqr | aqs | aqt | ars | art | ast | bqr | bqs | bqt | brs | brt | bst | qrs | qrt | qst | rst |

h | i | k | m | o | u | hik | him | hio | hiu | hkm | hko | hku | hmo | hmu | hou | ikm | iko | iku | imo | imu | iou | kmo | kmu | kou | mou |

g | j | l | n | o | u | gjl | gjn | gjo | gju | gln | glo | glu | gno | gnu | gou | jln | jlo | jlu | jno | jnu | jou | lno | lnu | lou | nou |

d | e | l | m | p | u | del | dem | dep | deu | dlm | dlp | dlu | dmp | dmu | dpu | elm | elp | elu | emp | emu | epu | lmp | lmu | lpu | mpu |

c | f | k | n | p | u | cfk | cfn | cfp | cfu | ckn | ckp | cku | cnp | cnu | cpu | fkn | fkp | fku | fnp | fnu | fpu | knp | knu | kpu | npu |

b | f | j | m | q | u | bfj | bfm | bfq | bfu | bjm | bjq | bju | bmq | bmu | bqu | fjm | fjq | fju | fmq | fmu | fqu | jmq | jmu | jqu | mqu |

a | e | i | n | q | u | aei | aen | aeq | aeu | ain | aiq | aiu | anq | anu | aqu | ein | eiq | eiu | enq | enu | equ | inq | inu | iqu | nqu |

a | c | g | m | r | u | acg | acm | acr | acu | agm | agr | agu | amr | amu | aru | cgm | cgr | cgu | cmr | cmu | cru | gmr | gmu | gru | mru |

b | d | h | n | r | u | bdh | bdn | bdr | bdu | bhn | bhr | bhu | bnr | bnu | bru | dhn | dhr | dhu | dnr | dnu | dru | hnr | hnu | hru | nru |

a | d | j | k | s | u | adj | adk | ads | adu | ajk | ajs | aju | aks | aku | asu | djk | djs | dju | dks | dku | dsu | jks | jku | jsu | ksu |

b | c | i | l | s | u | bci | bcl | bcs | bcu | bil | bis | biu | bls | blu | bsu | cil | cis | ciu | cls | clu | csu | ils | ilu | isu | lsu |

g | h | p | q | s | u | ghp | ghq | ghs | ghu | gpq | gps | gpu | gqs | gqu | gsu | hpq | hps | hpu | hqs | hqu | hsu | pqs | pqu | psu | qsu |

e | f | o | r | s | u | efo | efr | efs | efu | eor | eos | eou | ers | eru | esu | for | fos | fou | frs | fru | fsu | ors | oru | osu | rsu |

b | e | g | k | t | u | beg | bek | bet | beu | bgk | bgt | bgu | bkt | bku | btu | egk | egt | egu | ekt | eku | etu | gkt | gku | gtu | ktu |

a | f | h | l | t | u | afh | afl | aft | afu | ahl | aht | ahu | alt | alu | atu | fhl | fht | fhu | flt | flu | ftu | hlt | hlu | htu | ltu |

c | d | o | q | t | u | cdo | cdq | cdt | cdu | coq | cot | cou | cqt | cqu | ctu | doq | dot | dou | dqt | dqu | dtu | oqt | oqu | otu | qtu |

i | j | p | r | t | u | ijp | ijr | ijt | iju | ipr | ipt | ipu | irt | iru | itu | jpr | jpt | jpu | jrt | jru | jtu | prt | pru | ptu | rtu |

c | f | l | m | o | v | cfl | cfm | cfo | cfv | clm | clo | clv | cmo | cmv | cov | flm | flo | flv | fmo | fmv | fov | lmo | lmv | lov | mov |

d | e | k | n | o | v | dek | den | deo | dev | dkn | dko | dkv | dno | dnv | dov | ekn | eko | ekv | eno | env | eov | kno | knv | kov | nov |

g | j | k | m | p | v | gjk | gjm | gjp | gjv | gkm | gkp | gkv | gmp | gmv | gpv | jkm | jkp | jkv | jmp | jmv | jpv | kmp | kmv | kpv | mpv |

h | i | l | n | p | v | hil | hin | hip | hiv | hln | hlp | hlv | hnp | hnv | hpv | iln | ilp | ilv | inp | inv | ipv | lnp | lnv | lpv | npv |

a | d | h | m | q | v | adh | adm | adq | adv | ahm | ahq | ahv | amq | amv | aqv | dhm | dhq | dhv | dmq | dmv | dqv | hmq | hmv | hqv | mqv |

b | c | g | n | q | v | bcg | bcn | bcq | bcv | bgn | bgq | bgv | bnq | bnv | bqv | cgn | cgq | cgv | cnq | cnv | cqv | gnq | gnv | gqv | nqv |

b | e | i | m | r | v | bei | bem | ber | bev | bim | bir | biv | bmr | bmv | brv | eim | eir | eiv | emr | emv | erv | imr | imv | irv | mrv |

a | f | j | n | r | v | afj | afn | afr | afv | ajn | ajr | ajv | anr | anv | arv | fjn | fjr | fjv | fnr | fnv | frv | jnr | jnv | jrv | nrv |

b | f | h | k | s | v | bfh | bfk | bfs | bfv | bhk | bhs | bhv | bks | bkv | bsv | fhk | fhs | fhv | fks | fkv | fsv | hks | hkv | hsv | ksv |

a | e | g | l | s | v | aeg | ael | aes | aev | agl | ags | agv | als | alv | asv | egl | egs | egv | els | elv | esv | gls | glv | gsv | lsv |

i | j | o | q | s | v | ijo | ijq | ijs | ijv | ioq | ios | iov | iqs | iqv | isv | joq | jos | jov | jqs | jqv | jsv | oqs | oqv | osv | qsv |

c | d | p | r | s | v | cdp | cdr | cds | cdv | cpr | cps | cpv | crs | crv | csv | dpr | dps | dpv | drs | drv | dsv | prs | prv | psv | rsv |

a | c | i | k | t | v | aci | ack | act | acv | aik | ait | aiv | akt | akv | atv | cik | cit | civ | ckt | ckv | ctv | ikt | ikv | itv | ktv |

b | d | j | l | t | v | bdj | bdl | bdt | bdv | bjl | bjt | bjv | blt | blv | btv | djl | djt | djv | dlt | dlv | dtv | jlt | jlv | jtv | ltv |

e | f | p | q | t | v | efp | efq | eft | efv | epq | ept | epv | eqt | eqv | etv | fpq | fpt | fpv | fqt | fqv | ftv | pqt | pqv | ptv | qtv |

g | h | o | r | t | v | gho | ghr | ght | ghv | gor | got | gov | grt | grv | gtv | hor | hot | hov | hrt | hrv | htv | ort | orv | otv | rtv |

d | f | g | i | u | v | dfg | dfi | dfu | dfv | dgi | dgu | dgv | diu | div | duv | fgi | fgu | fgv | fiu | fiv | fuv | giu | giv | guv | iuv |

c | e | h | j | u | v | ceh | cej | ceu | cev | chj | chu | chv | cju | cjv | cuv | ehj | ehu | ehv | eju | ejv | euv | hju | hjv | huv | juv |

a | b | o | p | u | v | abo | abp | abu | abv | aop | aou | aov | apu | apv | auv | bop | bou | bov | bpu | bpv | buv | opu | opv | ouv | puv |

k | l | q | r | u | v | klq | klr | klu | klv | kqr | kqu | kqv | kru | krv | kuv | lqr | lqu | lqv | lru | lrv | luv | qru | qrv | quv | ruv |

m | n | s | t | u | v | mns | mnt | mnu | mnv | mst | msu | msv | mtu | mtv | muv | nst | nsu | nsv | ntu | ntv | nuv | stu | stv | suv | tuv |

Regards

Colin Fairbrother

]]>
with Multiple Wins

In the Lotto community that is interested in the structure of a set of numbers played the emphasis has always been on producing the minimum set of lines to ensure a single win for a given Lotto game. You don't have to be too long in this area of interest and doing some serious analysis to find that to be distortionary. My emphasis is looking at things from the perspective of Percentage Return or Yield over a reasonable number of draws for the set of numbers played and an honest realistic analysis free

as much as is possible of mathematical terminology.**To that end if considering a 6/49 Lotto game with a ticket cost of $0.50 and payout of $2.00 ****for a single three win**** we have to consider multiple lowest level prize wins to be ahead in anything more than 3 lines.**

as much as is possible of mathematical terminology.

A good starting point when considering multiple wins is to consider a Pool of 22 for a Pick or Selection of 6. Without any knowledge of Covers you would in any serious analysis of the Ones, Twos, Threes, Fours and Fives that are possible from this Pool of 22 become aware of some remarkable structural details. Most notable is that from the 20 threes that can be formed from a six, 77 lines can be constructed that contain the 1540 distinct threes ie they are packed into the minimum possible set of lines. (77x20=1540). In mathematical circles where they like to attribute increments of knowledge to someone or another this is known as a Steiner set after the Swiss Jakob Steiner around 1853. Usually, in Cover parlance this is written as C(22,6,3,3)=77 but more informative would be C(22,6,3,3,1)=77 where the 1 is not assumed and indicates the minimum prize win.

You can easily produce this Cover. An easy check to establish there are no duplicates is to test against the 22c3=1540 possible unique or distinct Threes. In my first attempt at constructing C(49,6,3,6)=365 with no duplicate subset Threes I tried rather than Lexicographic order Last Number order. Naturally, after completing Pool 22 I had the Steiner set of 77 lines; after 450 lines full coverage for 1 Three was obtained without any duplicate CombThrees.

With no computers in the 1850's you can try reproducing this cover. Noting you can't repeat a combination of 3 integers you

could start as follows lexicographically (numerical order) or increasing the Pool progressively (Last Number Order, LNO)

1 01 02 03 04 05 06 1 01 02 03 04 05 06

2 01 02 07 08 09 10 2 01 02 07 08 09 10

3 01 02 11 12 13 14 3 03 04 07 08 11 12

4 01 02 15 16 17 18 4 05 06 09 10 11 12

. . . . . .

. . . . . .

77 13 14 16 17 20 22 77 13 14 19 20 21 22

With no computers in the 1850's you can try reproducing this cover. Noting you can't repeat a combination of 3 integers you

could start as follows lexicographically (numerical order) or increasing the Pool progressively (Last Number Order, LNO)

1 01 02 03 04 05 06 1 01 02 03 04 05 06

2 01 02 07 08 09 10 2 01 02 07 08 09 10

3 01 02 11 12 13 14 3 03 04 07 08 11 12

4 01 02 15 16 17 18 4 05 06 09 10 11 12

. . . . . .

. . . . . .

77 13 14 16 17 20 22 77 13 14 19 20 21 22

Considering a hypothetical but very possible 6/22 Lotto game and the 7,315 possible CombFours then the minimum prize level is 4 CombThree wins. So the same 77 lines Cover could be written as C(22,6,3,4,4)=77. For the possible 26,334 CombFives then the minimum prize level becomes 10 Three wins - so, the same set could be described as C(22,6,3,5,10)=77. For the possible 13,983,816 CombSixes the minimum prize level is 20 CombThree Wins and so the same set could be written as C(22,6,3,6,20)=77. (The highest prize level, if you are very, very lucky, for one of the 77 lines could be the Jackpot)

An interesting exercise is to actually see when you would get these 2,464 blocks that give 20 Threes which occur in 3.3% of the 74,613 combinations of six that can be formed from a pool of 22. You can do this easily in Excel, Access, VB or C# by first enumerating the combinations then doing a find. If you constructed your Cover in lexicographic order the first should be at index 502 showing 1 2 3 7 17 20 - if so inclined you can find the 2,463 others. In passing I mention that 74,613 sixes is only around half of 1% of the 13,983,816 sixes that can be formed from a pool of 49.

The Percentage Return (Wins/Cost x 100) from this set of 77 Combs for the multiple wins is better in each case when compared to the best Cover for a single win but only the

C(22,6,3,6,20)=77 gives winnings in excess of the extra cost.

C(22,6,3,6,20)=77 gives winnings in excess of the extra cost.

Cover Percentage Return Profit $ Loss $

(if success) (failed)

(if success) (failed)

C(22,6,3,4,1)=46 2/23 x 100 = 8.7% (21.00) 23.00

C(22,6,3,4,4)=77 8/38.5 x 100 = 20.8% (30.50) 38.50

C(22,6,3,4,4)=77 8/38.5 x 100 = 20.8% (30.50) 38.50

C(22,6,3,5,1)=22 2/11 x 100 = 18.2% (9.00) 11.00

C(22,6,3,5,10)=77 20/38.5 x 100 = 51.9% (18.50) 38.50

C(22,6,3,6,1)=15 2/7.5 x 100 = 26.7% (5.50) 7.50

C(22,6,3,6,20)=77 40/38.5 x 100 = 103.9%** 1.50** 38.50

C(22,6,3,5,10)=77 20/38.5 x 100 = 51.9% (18.50) 38.50

C(22,6,3,6,1)=15 2/7.5 x 100 = 26.7% (5.50) 7.50

C(22,6,3,6,20)=77 40/38.5 x 100 = 103.9%

(For those who like to have a punt on the Gee Gees the last line is enough for them to ask - "Why is it a certainty?". Your embarrassed answer would be - "Well, it isn't! It's the government you see.")

Not all comparisons work out as well as our maximum threes compacted set. eg C(10,6,3,3,1)=10 and C(10,6,3,3,2)=18. **On a closer examination taking the cost of a ticket as being 50¢, a Three win as $2.00 and assuming you got 3 from the Pool of 10 right then for the 10 lines you are $3.00 out of pocket compared to $5.00 for the 18 lines!** The usual comparison made is that you save $1.00 over playing 2 x 10 lines with the same numbers in the same draw.

Regards

Colin Fairbrother

Colin Fairbrother

a1 | a2 | a3 | a4 | a5 | a6 |
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1 | 4 | 5 | 6 | 7 | 8 |

1 | 2 | 3 | 5 | 7 | 8 |

1 | 2 | 3 | 4 | 6 | 8 |

2 | 3 | 4 | 5 | 6 | 7 |

9 | 11 | 16 | 17 | 18 | 19 |

9 | 11 | 12 | 17 | 19 | 22 |

9 | 11 | 12 | 16 | 18 | 22 |

9 | 10 | 11 | 13 | 15 | 20 |

9 | 10 | 11 | 13 | 14 | 21 |

10 | 13 | 16 | 17 | 18 | 19 |

10 | 12 | 13 | 18 | 19 | 22 |

10 | 12 | 13 | 16 | 17 | 22 |

12 | 14 | 15 | 20 | 21 | 22 |

14 | 15 | 17 | 18 | 20 | 21 |

14 | 15 | 16 | 19 | 20 | 21 |

a1 | a2 | a3 | a4 | a5 | a6 |
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1 | 9 | 11 | 17 | 18 | 19 |

1 | 5 | 8 | 13 | 17 | 21 |

1 | 7 | 19 | 20 | 21 | 22 |

1 | 2 | 9 | 10 | 13 | 16 |

1 | 4 | 8 | 12 | 15 | 16 |

1 | 2 | 3 | 4 | 5 | 6 |

2 | 3 | 8 | 13 | 19 | 20 |

2 | 3 | 10 | 17 | 18 | 22 |

2 | 5 | 7 | 11 | 13 | 14 |

2 | 4 | 7 | 15 | 17 | 21 |

2 | 5 | 12 | 16 | 19 | 22 |

3 | 10 | 12 | 13 | 14 | 21 |

3 | 4 | 16 | 18 | 19 | 21 |

3 | 8 | 11 | 14 | 16 | 22 |

3 | 6 | 7 | 9 | 14 | 18 |

3 | 5 | 9 | 15 | 16 | 20 |

4 | 6 | 7 | 8 | 13 | 22 |

4 | 5 | 9 | 14 | 21 | 22 |

4 | 11 | 15 | 18 | 20 | 22 |

5 | 8 | 10 | 14 | 18 | 20 |

6 | 12 | 17 | 18 | 20 | 21 |

6 | 10 | 11 | 15 | 16 | 19 |

7 | 8 | 9 | 10 | 11 | 12 |

13 | 14 | 15 | 16 | 17 | 18 |

a1 | a2 | a3 | a4 | a5 | a6 |
---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 |

1 | 2 | 7 | 8 | 9 | 10 |

3 | 4 | 7 | 8 | 11 | 12 |

5 | 6 | 9 | 10 | 11 | 12 |

5 | 6 | 7 | 8 | 13 | 14 |

3 | 4 | 9 | 10 | 13 | 14 |

1 | 2 | 11 | 12 | 13 | 14 |

3 | 5 | 7 | 9 | 15 | 16 |

4 | 6 | 8 | 10 | 15 | 16 |

1 | 6 | 7 | 11 | 15 | 17 |

2 | 5 | 8 | 12 | 15 | 17 |

2 | 4 | 9 | 11 | 16 | 17 |

1 | 3 | 10 | 12 | 16 | 17 |

2 | 3 | 10 | 11 | 15 | 18 |

1 | 4 | 9 | 12 | 15 | 18 |

1 | 5 | 8 | 11 | 16 | 18 |

2 | 6 | 7 | 12 | 16 | 18 |

3 | 6 | 8 | 9 | 17 | 18 |

4 | 5 | 7 | 10 | 17 | 18 |

13 | 14 | 15 | 16 | 17 | 18 |

2 | 4 | 7 | 13 | 15 | 19 |

1 | 3 | 8 | 14 | 15 | 19 |

1 | 6 | 9 | 13 | 16 | 19 |

2 | 5 | 10 | 14 | 16 | 19 |

3 | 5 | 11 | 13 | 17 | 19 |

4 | 6 | 12 | 14 | 17 | 19 |

8 | 10 | 12 | 13 | 18 | 19 |

7 | 9 | 11 | 14 | 18 | 19 |

1 | 5 | 10 | 13 | 15 | 20 |

2 | 6 | 9 | 14 | 15 | 20 |

2 | 3 | 8 | 13 | 16 | 20 |

1 | 4 | 7 | 14 | 16 | 20 |

7 | 9 | 12 | 13 | 17 | 20 |

8 | 10 | 11 | 14 | 17 | 20 |

4 | 6 | 11 | 13 | 18 | 20 |

3 | 5 | 12 | 14 | 18 | 20 |

4 | 5 | 8 | 9 | 19 | 20 |

3 | 6 | 7 | 10 | 19 | 20 |

11 | 12 | 15 | 16 | 19 | 20 |

1 | 2 | 17 | 18 | 19 | 20 |

8 | 9 | 11 | 13 | 15 | 21 |

7 | 10 | 12 | 14 | 15 | 21 |

4 | 5 | 12 | 13 | 16 | 21 |

3 | 6 | 11 | 14 | 16 | 21 |

2 | 6 | 10 | 13 | 17 | 21 |

1 | 5 | 9 | 14 | 17 | 21 |

1 | 3 | 7 | 13 | 18 | 21 |

2 | 4 | 8 | 14 | 18 | 21 |

1 | 4 | 10 | 11 | 19 | 21 |

2 | 3 | 9 | 12 | 19 | 21 |

7 | 8 | 16 | 17 | 19 | 21 |

5 | 6 | 15 | 18 | 19 | 21 |

2 | 5 | 7 | 11 | 20 | 21 |

1 | 6 | 8 | 12 | 20 | 21 |

3 | 4 | 15 | 17 | 20 | 21 |

9 | 10 | 16 | 18 | 20 | 21 |

3 | 6 | 12 | 13 | 15 | 22 |

4 | 5 | 11 | 14 | 15 | 22 |

7 | 10 | 11 | 13 | 16 | 22 |

8 | 9 | 12 | 14 | 16 | 22 |

1 | 4 | 8 | 13 | 17 | 22 |

2 | 3 | 7 | 14 | 17 | 22 |

2 | 5 | 9 | 13 | 18 | 22 |

1 | 6 | 10 | 14 | 18 | 22 |

2 | 6 | 8 | 11 | 19 | 22 |

1 | 5 | 7 | 12 | 19 | 22 |

9 | 10 | 15 | 17 | 19 | 22 |

3 | 4 | 16 | 18 | 19 | 22 |

1 | 3 | 9 | 11 | 20 | 22 |

2 | 4 | 10 | 12 | 20 | 22 |

5 | 6 | 16 | 17 | 20 | 22 |

7 | 8 | 15 | 18 | 20 | 22 |

4 | 6 | 7 | 9 | 21 | 22 |

3 | 5 | 8 | 10 | 21 | 22 |

1 | 2 | 15 | 16 | 21 | 22 |

11 | 12 | 17 | 18 | 21 | 22 |

13 | 14 | 19 | 20 | 21 | 22 |