Discount Gambling

Wild Six Card Draw Poker

Posted in collusion by stephenhow on June 14, 2013

While I was playing Six Card Poker at my local Viejas Casino, another player told be about the Wild Six Card Draw that he plays in Colorado. It’s a poker game with two wild Jokers in the 54-card deck, and the player gets 5 cards plus one free replacement card vs the dealer’s 6 card hand. I ran a Monte Carlo analysis to see if ideal 6-way collusion would yield any edge (you never know, the game has two Jokers after all). But even with 6-way collusion, you can’t get the house edge below 2.2%. I guess that makes sense, since it’s probably rare where you’d chose a weird draw over the more obvious discard. Anyway, it’s really easy to check these things, and you never know what you’ll find.

Super Six Baccarat

Posted in +EV, baccarat by stephenhow on June 14, 2013

Some readers asked about a Baccarat side bet called “Super Six” which pays 15:1 for a dealer wins with a 6 total. It’s really easy to analyze the countability of any Baccarat side bet. The ideal return for this bet with a perfect (computer) count of an 8-deck shoe game with 15 cards behind the cut is only +24% of a fixed bet per shoe (2.6 bets per shoe at an average +9.2% advantage per bet). A simple unbalanced count (six => -2, seven, eight, nine => +1) and betting when the running count is +34 or higher yields only +12.2% of a fixed bet per shoe on 2.77 bets/shoe, and +4.41% edge/bet. It really doesn’t seem worth the effort, even if you had an ideal count (e.g., mobile app). You’d go crazy waiting around for less than 3 bets per shoe.

Edge Sorting Groups for Mississippi Stud

Posted in +EV, edge sorting, mississippi stud by stephenhow on June 13, 2013

Mississippi Stud LogoYou probably know that I’m not much into advantage play based on edge-sorting cards. That’s the realm of Phil Ivey and Eliot Jacobson. It’s a pretty cool technique, but it’s way too involved for my attention span, regardless of the payoff. However, I did watch Warren Beatty in Kaleidescope, if that counts for anything.

Anyways, a reader who saw Eliot’s post on Edge Sorting (Jacks in) Mississippi Stud asked me if it’d be worthwhile to also sort the Queens, Kings, and Aces. That’s a pretty interesting question, since I can see how Eliot would start out with just the Jacks, as you’d know when you had a sure winner. But, maybe sorting the other “pay” cards would improve the return. You might not know exactly when you had a winner, but you’d have a good idea, and much more often.

I realised a Monte Carlo analysis would easily yield the ideal return for any selected sorting group. I modified a few lines of code, and violá, I simulated the estimated theoretical max return for the following sorted card groups in Mississippi Stud:

Max Return for Known Card Groups
Sorted Card Group Ideal Return
Jacks +39.7%
Jacks & Queens +48.9%
Jacks, Queens, Kings +59.0%
Jacks, Queens, Kings, Aces +63.4%

(I use the paytable that pays 5:1 for a straight.)

So it’s probably worthwhile to sort all the “pay” cards, unless it really complicates the practical strategy (not too likely).

While it’s easy to get the return for an ideal strategy for any sorting group, it takes time to work out a practical strategy. It’s straightforward, but tedious, so I’m not doing it. (Well, I actually did it for a reader, so it’s his now.)

Lucky Stiff BJ Side Bet @ 7 Cedars, WA

Posted in blackjack sidebets by stephenhow on May 28, 2013

luckystiffPaigow Dan told me about the new Lucky Stiff side bet his friend recently placed at the 7 Cedars Casino in WA. It looks fun, because you’re paid 5:1 when your initial 12-16 hard total ends up winning the main hand. Also, blackjack pays even-money on the side bet, and an initial pair of 8-8, 7-7, and 6-6 instantly wins 10:1. Anyways, I ran the bet through my BJ analyzer, to see if it was interesting in any way. I understand that 7 Cedars lets you bet $5 on the main hand, and up to $25 on the side bet. So I ran the analysis for a 5:1 side-to-main ratio on a 6-deck, H17, SP4, SPA4 game. The return showed a house edge of 3.5% of the combined (main+side) wager. The optimal strategy for the 5:1 side-to-main ratio only has a few differences with basic strategy.

Hand Dealer Upcard
2 3 4 5 6 7 8 9 10 A
Soft Totals
soft 21 S S S S S S S S S S
soft 20 S S S S S S S S S S
soft 19 S S S S D S S S S S
soft 18 D D D D D S S H H H
soft 17 H D D D D H H H H H
soft 16 H H D D D H H H H H
soft 15 H H D D D H H H H H
soft 14 H H H D D H H H H H
soft 13 H H H D D H H H H H
Hard Totals
hard 20 S S S S S S S S S S
hard 19 S S S S S S S S S S
hard 18 S S S S S S S S S S
hard 17 S S S S S S S S S S
hard 16 S S S S S H H H S H
hard 15 S S S S S H H H S H
hard 14 S S S S S H H H H H
hard 13 S S S S S H H H H H
hard 12 H S S S S H H H H H
hard 11 D D D D D D D D D D
hard 10 D D D D D D D D H H
hard 9 H D D D D H H H H H
hard 8 H H H H H H H H H H
hard 7 H H H H H H H H H H
hard 6 H H H H H H H H H H
hard 5 H H H H H H H H H H
Pairs
A-A P P P P P P P P P P
10-10 S S S S S S S S S S
9-9 P P P P P S P P S S
8-8 P P P P P P P P P P
7-7 P P P P P P H H H H
6-6 P P P P P H H H H H
5-5 D D D D D D D D H H
4-4 H H H P P H H H H H
3-3 P P P P P P H H H H
2-2 P P P P P P H H H H

The EORs are fairly small for the 5:1 side-to-main ratio. They’re about only 1/3rd as effective as the EORs for a standard 6-deck shoe main game. So it’s not worth your time to count this side bet. For a single card removed in a 6-deck game, the EORs are as follows:

Lucky Stiff EORs (5:1 side-to-main ratio, 6 decks)
Card Removed Return EOR
None 3.5009%
Ace 3.1324% 0.3685%
Deuce 3.3055% 0.1954%
Trey 3.3723% 0.1285%
Four 3.4423% 0.0585%
Five 3.5247% -0.0238%
Six 4.0443% -0.5434%
Seven 3.7864% -0.2856%
Eight 3.7654% -0.2845%
Nine 3.3903% 0.1106%
Ten/Face 3.4194% 0.0815%

This bet looks like fun. If you bet an equal main and side bet (1:1 side-to-main ratio), the house edge is 4.66% on the combined 2 unit bet (2.33% element-of-risk). That’s not too bad for a carnival-like odds. If you make a small side bet 1/5th of your main bet (e.g., a $1 side bet to a $5 main bet), then the house edge on the combined 1.2 unit bet is 1.38%. That’s not bad for a little bit of fun.

Suit’Em Up BJ Side Bet @ Venetian, LV

Posted in +EV, blackjack sidebets by stephenhow on May 16, 2013

I saw this blackjack side bet in the Venetian last month, and it looked pretty you-know-what. I forgot to post about it until now. I’m pretty sure they use 8-deck shoes at the Venetian.

Suit’Em Up BJ Side Bet (8 Decks)
Hand Combinations Frequency Payout Return
Suited Aces 112 0.001297 60 0.077850
Suited BJs 1,024 0.011863 10 0.118628
Suited Pairs 1,344 0.015570 5 0.077850
Suited 11’s 1,024 0.011863 3 0.035589
Other Suited 17,920 0.207560 2 0.415199
nothing 64,896 0.751807 -1 -0.751807
total 86,320 1.000000 -0.026691
Suit’Em Up EORs (8 Decks)
Removed Card EOR Balanced Count Unbalanced Count
Deuce +0.000767 +1 +1
Trey +0.000767 +1 +1
Four +0.000767 +1 +1
Five +0.000767 +1 +1
Six +0.000767 +1 +1
Seven +0.000767 +1 +1
Eight +0.000767 +1 +1
Nine +0.000767 +1 +1
Ten +0.000116 0 +1
Jack +0.000116 0 0
Queen +0.000116 0 0
King +0.000116 0 0
Ace -0.006601 -8 -8

Using the unbalanced taps, the bet is +EV for RC >= +34 (assuming two decks behind the cut card). This yields 16% betting opportunities, with an average edge of +2.8%/bet. The theoretical max (using full shoe composition, including suits) is 17% opportunities @ +3.0%/bet. It’s not worth much.

Bust Bonus BJ Side Bet

Posted in blackjack sidebets by stephenhow on May 14, 2013

BustBonus-BannerA reader pointed me out to Galaxy Gaming’s Bust Bonus blackjack side bet that the dealer will bust, which you make after seeing the dealer’s upcard. I figured I’d run the numbers to see if it was any good, or if it was countable. Well, it might be a fun bet on a few upcards, but it’s kind of expensive for the (offsuit) odds they offer.

Bust Bonus Side Bet (6 Decks)
Dealer
Upcard
Probability
Suited Bust
Probability
Offsuit Bust
Probability
No Bust
Payout
Suited Bust
Payout
Offsuit Bust
Return
Ace* 0.002222 0.199059 0.798719 50 3 -0.090437
Deuce 0.008751 0.347909 0.643339 25 1 -0.076644
Trey 0.011524 0.365435 0.623042 15 1 -0.084754
Four 0.014574 0.383896 0.601530 10 1 -0.071892
Five 0.017883 0.401749 0.580368 5 1 -0.089206
Six 0.020844 0.418415 0.560741 3 1 -0.079793
Seven 0.011716 0.250219 0.738064 15 2 -0.061881
Eight
888
0.011133
0.000207
0.227312
0.005041
0.756307
 
10
75
2
25
-0.048772
Nine 0.011602 0.217640 0.770758 20 2 -0.103441
Ten/Face* 0.012263 0.217975 0.769761 20 2 -0.088547

*Bust Bonus wagered after dealer peeks for blackjack.

The most countable bet is against a dealer 8 upcard. It has the lowest house edge (4.9%), and has high payouts for the 888o and 888s busts. The EORs are large, and a simple unbalanced count (Eight => -8, Nine, Ten/Face, Ace, Deuce, Trey, Four => +1; bet when running count >= +24) yields an average +7.5% edge/bet on 17.3% of the dealer 8 upcard hands. Of course, a dealer 8 only occurs on 1/13th of the hands, so it’s not a very practical bet. An ideal count (using total shoe composition including suits) yields a theoretical max return of +7.5% edge/bet on 1.6% of the dealt hands.

Sadly, the standard BJ counts (like the unbalanced Knockout count) don’t correlate with the EV of any of these bets, because unlike blackjack, the Ace hurts the Bust Bonus bet. (Ace rich shoe makes it harder to bust.)

High Card Flush

Posted in +EV, collusion by stephenhow on April 22, 2013

GGsite_HCF_ProdBanner_041713A couple of readers have asked about Galaxy Gaming’s new High Card Flush game, which has a few placements now, and may be picking up some steam. The game is pretty simple, where each player and the dealer receive 7 cards. Each hand is measured by its highest flush, where a flush is first ranked by its length (number of cards of same suit), then by its card values. Each player must Ante before the hand, then wagers a 1x-3x Play bet (depending on flush size), or folds. The dealer qualifies with a three-card, 9-high flush. If the dealer doesn’t qualify, the Play bets push, and the remaining Antes are paid even-money. If the dealer qualifies, the Ante and Play bets receive even-money action against the dealer hand.

As you would expect, collusion helps in this game. A Monte Carlo analysis shows that with 6 confederates, perfect knowledge of the dealt cards gives each spot at least a +7.3% edge over the house. But practically, you’d be lucky if you could even communicate the suit counts (number of cards of each suit) dealt. If you figure out a non-suspicious way of doing this, then the following simple strategy yields a +3.1% edge over the house:

High Card Flush Advanced Strategy (6 Player Hands Seen)
Flush Size Play Bet
1 or 2 cards 1x for suit counts (9, 11, 11, 11) or (10, 10, 11, 11), else
fold others
3 card, Jack-high or lower 1x for suit counts (9, 11, 11, 11) or (10, 10, 11, 11), else
fold others
3 card, Queen-high 1x if lowest suit count is 9 or higher,
else fold.
3 card, King-high or better 1x if lowest suit count is 8 or higher, else fold.
4 cards 1x
5 cards 2x
6 or 7 cards 3x

where the suit counts 4-tuple is the sorted number of cards of each suit.

Phil’Em Up Poker

Posted in collusion by stephenhow on April 19, 2013

When I playing Mississippi Stud in Vegas last week, I overheard someone mention a game called Phil’Em Up Poker. I looked at the game, to see if collusion would yield an edge. The rules are pretty simple. The game is played with a 52-card deck plus a Joker which may be used for Aces, straights, and flushes. Each player bets an Ante, and receives two hole cards. Two community cards are dealt face up. Each player may either make an additional 1x bet (i.e., “double-up” his action), or check, before the 3rd community card is exposed. If a player makes a pair of Tens or better, he wins according to a paytable. There is no dealer hand. The house edge is a reasonable 3.3%.

Optimal Outcomes for Phil’Em Up Poker
Hand Bet Combinations Probability Payout Return
FIVE_ACES 2 5 0.00000035 1000 0.000697
natural ROYAL_FLUSH 2 20 0.00000139 250 0.000697
wild ROYAL_FLUSH 2 100 0.00000697 100 0.001394
natural STRAIGHT_FLUSH 2 180 0.00001254 50 0.001254
wild STRAIGHT_FLUSH 2 720 0.00005018 25 0.002509
FOUR_OF_A_KIND 2 4,140 0.00028853 20 0.011541
FULL_HOUSE 2 21,840 0.00152212 15 0.045664
FLUSH 2 39,020 0.00271946 9 0.048950
STRAIGHT 2 77,460 0.00539850 7 0.075579
THREE_OF_A_KIND 2 211,200 0.01471939 3 0.088316
TWO_PAIRS 2 365,640 0.02548294 2 0.101932
High Pair 2 1,562,112 0.10886993 1 0.217740
Low Pair 2 75,648 0.00527222 -1 -0.010544
HIGH_CARD 2 339,708 0.02367563 -1 -0.047351
STRAIGHT 1 25,200 0.00175629 7 0.012294
THREE_OF_A_KIND 1 105,600 0.00735969 3 0.022079
TWO_PAIRS 1 327,360 0.02281505 2 0.045630
High Pair 1 922,608 0.06430030 1 0.064300
Low Pair 1 3,514,752 0.24495734 -1 -0.244957
HIGH_CARD 1 6,755,112 0.47079118 -1 -0.470791
total 14,348,425 -0.033067
expected 14,348,425

Collusion doesn’t help. That’s because only 3.8% of hands are bet on a draw only. Collusion will change few decisions, and result in little gain. With 7-player collusion, perfect play will only reduce the house edge to 3.2%.

Double Attack Blackjack

Posted in +EV, counting, double attack blackjack by stephenhow on March 29, 2013

wagerworks-double-attackThanks to reader John A. for pointing out this game to me. The game has been around (mostly in Atlantic City), but it’s new to me. It looks like the predecessor to Triple Attack Blackjack, as it’s based on a Spanish deck (10’s removed, J/Q/K’s remain) and the player may double his bet after the first card is dealt face up to the dealer. After this initial double attack option, the hand plays out normally with the total amount bet as the hand wager. (I.e., doubles and splits are based on the total amount bet after any double attack.)

The rules following the double-attack option are as follows:

  • Dealer stands on soft-17
  • Double-down at any time (no re-doubles)
  • Surrender at any time, including double-down rescue and after splits
  • No re-splitting of Aces
  • Blackjack pays even money

The house edge for the game is a reasonable 0.50% on the initial bet. The element-of-risk is even lower, as you double your wager 58% of the time (i.e., you double-attack vs. a dealer 2-8). The return is even lower still if they allow you to surrender after splitting Aces. The EORs are listed in the following table for removing a single card from a 8-deck shoe.

Double Attack Blackjack EORs (8 deck)
Removed Card EOR Balanced Unbalanced
Deuce +0.0832% +1 +1
Trey +0.1127% +1 +1
Four +0.1514% +1 +1
Five +0.1917% +1 +1
Six +0.1184% +1 +1
Seven +0.0341% +1
Eight -0.0560%
Nine -0.0895% -1 -1
Face -0.1466% -1 -1
Ace -0.0937% -1 -1

Basic Strategy

The basic strategy for the game was auto-generated by my analyzer program. You should double-down rescue 16 and lower against a dealer 8-thru-A, and 17 against an Ace. The strategy simulates at a -0.53% return, averaged over the whole shoe, very close to the analyzer’s calculated -0.50% return.

The unbalanced count in the above table yields 23.8% +EV betting opportunities (count >= +23) in an 6-deck shoe game with 52 cards behind the cut card. The average +EV hand returns +0.52%/bet. Compare this to the “Knockout” unbalanced count for 6-deck standard blackjack with cut card @ 5th deck, where 21.3% of the hands are +EV (count >= +17) with an average yield of +0.30%/bet.

Hand Dealer Upcard
2 3 4 5 6 7 8 9 10 A
Soft Totals
soft 21 S S S S S S S S S S
soft 20 S S S S S S S S S S
soft 19 S S S S S S S S S S
soft 18 S S S D D S S H H H
soft 17 H H D D D H H H H H
soft 16 H H H H D H H H H H
soft 15 H H H H H H H H H H
soft 14 H H H H H H H H H H
soft 13 H H H H H H H H H H
Hard Totals
hard 20 S S S S S S S S S S
hard 19 S S S S S S S S S S
hard 18 S S S S S S S S S S
hard 17 S S S S S S S S S R
hard 16 S S S S S H H H H H
hard 15 S S S S S H H H H H
hard 14 H H S S S H H H H H
hard 13 H H H H H H H H H H
hard 12 H H H H H H H H H H
hard 11 D D D D D D D D H D
hard 10 D D D D D D D H H H
hard 9 H H H H H H H H H H
hard 8 H H H H H H H H H H
hard 7 H H H H H H H H H H
hard 6 H H H H H H H H H H
hard 5 H H H H H H H H H H
Pairs
A-A P P P P P P P P P H
10-10 S S S S S S S S S S
9-9 S S P P P S P P S S
8-8 P P P P P P P P P P
7-7 P P P P P P H H H H
6-6 H H P P P H H H H H
5-5 D D D D D D D H H H
4-4 H H H H H H H H H H
3-3 H P P P P P P H H H
2-2 H P P P P P H H H H

Blackjack Switch

Posted in blackjack switch by stephenhow on March 24, 2013

bjswitch2Ok, I just spent way too much time working out a Blackjack Switch strategy. At first, I just wanted to calculate the EORs for Blackjack Switch, to see if it was more countable than regular blackjack. (The EORs are about the same as regular blackjack, so it’s probably not more countable.) Then I got carried away making a switching strategy, trying to keep it simple and intuitive (i.e., real). Hopefully, this post will save someone the bother of going through all this again.

House Edge

I put the Blackjack Switch rules in my blackjack analyzer program, which found the best switch decision and combined EV for each 4-card starting hand (100x more hands than standard BJ). I calculated the value of each 2-card hand assuming the other 2-card hand hadn’t yet played out (a simplifying assumption to make calculations practical). For a 6-deck shoe and Las Vegas rules (H17, DAS, re-split all pairs up to 4 hands, no LS, no BJ after switch), I got a combined return of -1.00% for both hands, which is a house edge of 0.50% per hand.

Effect of Removed Cards (EORs)

The sensitivity of the game’s EV to the removal of a given card rank is called the “effect of removal” (EOR). If Blackjack Switch was highly vulnerable to counting, you’d see it in the EORs. This is the first place to look. The table below lists the effect on the optimal EV of the game by removing one card of a given rank from a 6-deck shoe.

6-Deck Blackjack Switch EORs
Removed Rank EOR Balanced Unbalanced
Deuce 0.0959% +1 +1
Trey 0.0711% +1 +1
Four 0.0891% +1 +1
Five 0.1086% +1 +1
Six 0.1077% +1 +1
Seven 0.0379%
Eight 0.0166%
Nine 0.0564% -0.5
Ten/Face 0.0925% -1 -1
Ace 0.0561% -0.5

The EORs are very similar to those of regular blackjack. The sum of the 2-6 EORs is about 0.45% in both cases. However, the Ace is half as powerful compared to regular blackjack. In Blackjack Switch, an Ace plus a Nine is comparable to an Ace in regular blackjack. You could probably use your normal counting system for BJ Switch, but note that it’s overestimating the power of Aces, and underestimating the power of Nines.

I also broke out separate EORs for the switch EV (the nominal 9.25% advantage obtained through the player switch option). If these values were large, it’d indicate an exploit through an indexed switch strategy. However, these switch EORs are very low, about 5x lower than the overall EORs. So an indexed switching strategy would not yield much benefit.

Basic Strategy

Basic strategy for Blackjack switch consists of the initial switch decision, and the post-switch basic strategy table.

Switching Strategy

Blackjack Switch has a pretty big following, and probably no one follows any published strategy. I’ve played about 6 hands of this game IRL, and when I hit my 12 against a dealer deuce upcard, the other player at the table repeatedly pleaded with me, asking (rhetorically) “Why would you hit that?!”.

Judging by the game’s popularity, people don’t have any problems making their switch decisions. The analysis shows a handful of intuitive rules returns almost all the switch advantage, and is suboptimal by only 0.13% (per hand). The table below summarizes the prioritised switching strategy, with the frequencies and costs for each rule for a 6-deck shoe.

Simple Blackjack Switch Strategy
Rule Frequency Cost
Switch doesn’t change hands, else 26.94% 0%
Switch improves both hands1, else 28.60% 0.021%
Play desired2 hand(s) over no desired hands, else 20.03% 0.003%
vs. 2-6 dealer upcard
Play double and split, else 0.19% 0.004%
Maximize desired hand, else 7.90% 0.038%
ignore others 0.52% 0.002%
vs. 7-A dealer upcard
Play two strong3 hands against upcard, else 1.05% 0.002%
Play two non-weak4 hands over any weak5 hand(s), else 0.95% 0.004%
Play strong hand over non-strong hand, else 4.19% 0.005%
Play strong hand and non-bustable6 hand, else 1.20% 0.021%
Maximize desired hand if alternative is weak, else 4.72% 0.017%
Play 7/17 if no desired hand, else 0.97% 0.013%
ignore others 2.75% 0.011%
Total 100% 0.13%

1Or improves one hand without hurting other. See hand rankings defined below.
2Desired hands are defined by Cindy Liu, as (in descending order): BJ, 21, 20, 19, AA, 11, 10, 9, 8/18, and 8-8 vs. 2-6 upcard.
3Strong hand = desired hand with last digit of total greater than dealer upcard. For splits, use split card value.
4Non-Weak hand = desired hand with last digit of total greater than or equal to dealer upcard.
5Weak hand = desired hand with last digit of total less than dealer upcard.
6Non-bustable = stand, or hand that won’t bust on next card (splits, totals <= 11, soft-totals).

The hand ranking used for comparing two candidate hands against a fixed upcard is as follows. Hands in the same level are equal, except for sub-ranks in parenthesis.

  1. Desired hand. (Compare two desired hands by their rank.)
  2. Split hand.
  3. Any soft total.
  4. any hard total <= 7
  5. 17 vs. 2-6
  6. 12+ hitting hard total (lower is better)
  7. standing hand

Hopefully, the switching strategy is intuitive enough to understand without any detailed description of the rules. I’ve posted a whole bunch of example hands of the switching strategy that should clarify how it works in practice.

Post-Switch Strategy

Here’s the basic strategy for playing your hand after the switch. The strategy is auto-generated by my blackjack analyzer program for a 6-deck shoe game.

Hand Dealer Upcard
2 3 4 5 6 7 8 9 10 A
Soft Totals
soft 21 S S S S S S S S S S
soft 20 S S S S S S S S S S
soft 19 S S S S S S S S S S
soft 18 S S S D D S S H H H
soft 17 H H H D D H H H H H
soft 16 H H H H D H H H H H
soft 15 H H H H H H H H H H
soft 14 H H H H H H H H H H
soft 13 H H H H H H H H H H
Hard Totals
hard 20 S S S S S S S S S S
hard 19 S S S S S S S S S S
hard 18 S S S S S S S S S S
hard 17 S S S S S S S S S S
hard 16 S S S S S H H H H H
hard 15 S S S S S H H H H H
hard 14 S S S S S H H H H H
hard 13 H S S S S H H H H H
hard 12 H H H H S H H H H H
hard 11 D D D D D D D D H H
hard 10 D D D D D D D H H H
hard 9 H H H D D H H H H H
hard 8 H H H H H H H H H H
hard 7 H H H H H H H H H H
hard 6 H H H H H H H H H H
hard 5 H H H H H H H H H H
Pairs
A-A P P P P P P P P P P
10-10 S S S S S S S S S S
9-9 S S P P P S P P S S
8-8 P P P P P P P P H H
7-7 S P P P P P H H H H
6-6 H H P P P H H H H H
5-5 D D D D D D D H H H
4-4 H H H H H H H H H H
3-3 H H H P P P H H H H
2-2 H H H P P P H H H H