Vertical Melds for Poker.
Version of Monday 16 February 2015.
Dave Barber's other pages.

This report introduces some new combinations of cards for comparing Poker hands.

In most poker play, and in many other card games, the ranks of the cards are:

table one
RankSymbol
highestAceA
KingK
QueenQ
JackJ
TenT
Nine9
Eight8
Seven7
Six6
Five5
Four4
Three3
lowestTwo2

A common practice is to allow the Ace to be either high (above the King) or low (below the Two) at the player's discrection, but for simplicity we assume that the Ace will always be high.

Here are two new terms for old ideas:

• A horizontal meld is several cards of the equal rank, such as QQQ.
• A vertical meld is several cards in rank sequence, such as 7654.
The word meld, rare in Poker, is borrowed from the Rummy and Pinochle families, where it is ubiquitous. We retain flush for a meld of cards all of the same suit.

The gist of the system proposed here is not complicated. To determine who has the best hand in a Poker showdown, players compare their cards according to the two-step rule:

• Step 1:
• Whichever hand has the longest vertical meld (in other words, sequence) wins.
• If players' longest sequences are equal in length, the second-longest sequences are compared, then the third-longest, et cetera.
• The ranks of cards do not matter in step 1, except to the extent that sequences are formed. For instance, 9876 and 6543 are equally good.
• Step 2: If hands are still tied after step 1,
• Comparison is based on the highest card in the longest sequence in each hand.
• If there is still a tie, comparison goes to the highest card in the second-longest sequence in each hand; then if necessary to the highest card in the third-longest sequence in each hand; et cetera.
• An absolute tie is possible, but rare.

To illustrate, a player who holds Q-J-T-T-9 would regard the melds as QJT9 and T, not QJT and T9.

Examples are given in table two below. The greater-than symbol > stands for "defeats", and the small x stands for any irrelevant card. The hyphen is inserted for convenience of reading, separating sequences within the hand.

table two
combinations of vertical melds
assuming each player holds five cards
categorydescriptionexamples
highestlowestcomparisons
highest v5 • one sequence of five cards • AKQJT• 65432 • JT987 > T9876
v41 • one sequence of four cards
• one leftover card
• AKQJ-A• 5432-2 • 8765-J > 7654-K because 8765-x > 7654-x
• 7654-K > 7654-Q
v32 • one sequence of three cards
• one sequence of two cards
• AKQ-AK• 432-32 • 876-JT > 765-KQ because 876-xx > 765-xx
• 765-KQ > 765-QJ
• 765-76 > 765-65
v311 • one sequence of three cards
• two leftover cards
• AKQ-A-A• 432-2-2 • 876-J-3 > 765-K-J because 876-x-x > 765-x-x
• 765-K-7 > 765-Q-T because 765-K-x > 765-Q-x
• 765-Q-T > 765-Q-9
v221 • two sequences of two cards
• one leftover card
• AK-AK-A• 32-32-2 • 98-32-K > 87-54-A because 98-xx-x > 87-xx-x
• 87-54-2 > 87-43-K because 87-54-x > 87-43-x
• 87-87-2 > 87-54-K because 87-87-x > 87-54-x
• 87-43-K > 87-43-Q
v2111 • one sequence of two cards
• three leftover cards
• AK-A-A-A• 32-2-2-2 • 98-Q-6-2 > 87-K-7-5 because 98-x-x-x > 87-x-x-x
• 98-Q-6-2 > 98-J-5-3 because 98-Q-x-x > 98-J-x-x
• 98-Q-6-2 > 98-Q-5-3 because 98-Q-6-x > 98-Q-5-x
• 98-Q-6-3 > 98-Q-6-2
lowest v11111 • five leftover cards • A-A-A-A-A• 2-2-2-2-2 • K-J-8-6-4 > Q-Q-Q-9-4 because K-x-x-x-x > Q-x-x-x-x
• Q-Q-8-6-4 > Q-J-9-7-4 because Q-Q-x-x-x > Q-J-x-x-x
• Q-Q-Q-6-4 > Q-Q-9-7-4 because Q-Q-Q-x-x > Q-Q-9-x-x
• Q-Q-8-6-4 > Q-Q-8-4-2 because Q-Q-8-6-x > Q-Q-8-4-x
• Q-Q-8-6-4 > Q-Q-8-6-3

Note for instance that 432-76 defeats AKQ-A-A because any v32 defeats any v311. The question of which hand has cards of higher ranks arises only when two hands of the same category are compared, and the category is established in step one of the two-step rule.

Within the hand, one sequence will sometimes be contained within another (as QJT-JT or 54-54-9) or not contained (as QJT-43 or KQ-54-9). However, this makes no difference in the ranking of hands as proposed here. Of course, some players might choose to give contained sequences special treatment.

Although we do not treat horizontal and flush melds extensively, it is still worthwhile to establish notations.

table three
combinations of horizontal melds
assuming a pack with four cards per rank
categorydescriptionexample
highest h41 • quartet
• one leftover card
• 6666-8 > 5555-9
h32 • trio
• pair
• QQQ-33 > TTT-99
h311 • trio
• two leftover cards
• 888-7-6 > 777-A-T
h221 • two pairs
• one leftover card
• JJ-66-4 > 99-88-6
h2111 • one pair
• three leftover cards
• 33-5-4-2 > 22-J-T-8
lowest h11111 • five leftover cards • 8-6-5-4-3 > 7-6-5-4-3

Flushes are based on matching suit, not rank. In most Poker play, no suit is regarded as superior to any other, and traditionally the only flush meld recognized is the f5. In the flush generalizations of table four, the suits are spades (♠), clubs (♣), hearts (♥), and diamonds (♦). Although many European packs use other suit symbols, nearly all have four suits.

table four
combinations of flush melds
assuming a pack with four cards per rank, all of different suits
categorydescriptionexample
highest f5 • quintet • ♣♣♣♣♣
f41 • quartet
• one leftover card
• ♦♦♦♦-♠
f32 • trio
• pair
• ♥♥♥-♣♣
f311 • trio
• two leftover cards
• ♠♠♠-♦-♣
f221 • two pairs
• one leftover card
• ♣♣-♥♥-♠
lowest f2111 • one pair
• three leftover cards
• ♦♦-♣-♥-♠

For similar observations on dice, see Permucolor.

For illustration, table five shows the traditional Poker hands in our notation:

table five
standard Poker combinations
assuming a pack with four cards per rank, all of different suits
categorydescription
highest v5 and f5straight flush
h41four of a kind
f32full house
f5flush
v5straight
h311three of a kind
h221two pair
h2111one pair
lowest h11111high card

By long tradition, a Poker hand has five cards. Although five-suit packs of cards have been published, they have for unclear reasons rarely been adopted by Poker players, even though a meld with five cards of the same rank would be possible, as well as a hand with five different suits.

Returning to vertical melds, table six tells how many hands fall into each of the seven categories; displayed are both a count and a percentage. Although the 13-rank pack is the most common in the United States and Britain, packs of 12 or 10 ranks are common in southern Europe, and 8 ranks in central Europe. For 14 ranks, players can use a Rook pack or the minor arcana from a Tarot pack; and to obtain 15 ranks, players can use 60 cards from a six-handed 500 pack.

In general, reducing the number of ranks leads to an increase in the occurence of longer sequences. Reducing the number of suits, while perfectly feasible, is almost never done. See also stripped deck.

table six
distribution of combinations of melds by category
assuming each player holds five cards,
and the pack has four cards per rank
15 ranks 13 ranks 12 ranks 10 ranks 8 ranks
v5 11,264 0.21% 9,216 0.35% 8,192 0.48% 6,144 0.93% 4,096 2.03%
v41 131,072 2.40% 89,088 3.43% 71,168 4.16% 41,472 6.30% 19,968 9.92%
v32 116,384 2.13% 76,896 2.96% 60,224 3.52% 33,024 5.02% 14,016 6.96%
v311 714,000 13.07% 399,984 15.39% 285,984 16.70% 128,640 19.55% 45,024 22.36%
v221 631,392 11.56% 343,584 13.22% 240,816 14.06% 102,192 15.53% 32,304 16.04%
v2111 2,440,432 44.68% 1,121,952 43.17% 717,080 41.88% 249,672 37.94% 64,632 32.10%
v1111 1,416,968 25.94% 558,240 21.48% 328,840 19.20% 96,864 14.72% 21,336 10.60%
total 5,461,512100.00% 2,598,960100.00% 1,712,304100.00% 658,008100.00% 201,376100.00%

Such a table does not tell the whole story in Draw Poker, because players can discard unsatisfactory cards and draw replacements that are potentially better. Under the ranking of the vertical melds as given above, a player has very little to lose by discarding a "leftover" card, in other words any card that forms a sequence whose length is only one. For example, with the category v311 hand 765-9-3, the 9 and 3 may be discarded with minimal risk, as their replacements cannot cannot drop the hand into v221 or any lower category. There is a small chance that, after the draw, the hand may turn out to be slightly weaker in an extended tiebreaker, as when 987-5-3 deteriorates to 987-4-2.

Table seven shows the probabilities with a pack of 13 ranks, and varying numbers of cards per rank. The percentages do not change much.

table seven
distribution of combinations of melds by category
assuming each player holds five cards,
and the pack has thirteen ranks
3 cards per rank 4 cards per rank 5 cards per rank 6 cards per rank
v5 2,187 0.38% 9,216 0.35% 28,125 0.34% 69,984 0.33%
v41 20,736 3.60% 89,088 3.43% 275,000 3.33% 689,472 3.27%
v32 18,090 3.14% 76,896 2.96% 236,000 2.86% 589,572 2.79%
v311 90,990 15.80% 399,984 15.39% 1,251,250 15.15% 3,164,562 14.99%
v221 78,858 13.70% 343,584 13.22% 1,069,900 12.95% 2,698,452 12.78%
v2111 248,022 43.08% 1,121,952 43.17% 3,568,100 43.20% 9,121,464 43.21%
v1111 116,874 20.30% 558,240 21.48% 1,831,513 22.17% 4,777,584 22.63%
total 575,757100.00% 2,598,960100.00% 8,259,888100.00% 21,111,090100.00%

Poker can also be played with a 48-card Pinochle pack, with only six ranks (A, K, Q, J, T, 9) but eight cards in each rank. Further, the nines can be removed from the pack for a tauter game. The prevalence of longer sequences is greatly increased, as enumerated in table eight.

table eight
distribution by category
assuming each player holds five cards,
and the pack has eight cards per rank
6 ranks 5 ranks
v5 65,536 3.83% 32,768 4.98%
v41 237,568 13.87% 114,688 17.43%
v32 115,712 6.76% 37,632 5.72%
v311 412,160 24.07% 165,760 25.19%
v221 262,976 15.36% 107,520 16.34%
v2111 457,184 26.70% 144,256 21.92%
v1111 161,168 9.41% 55,384 8.42%
total 1,712,304100.00% 658,008100.00%

Among most players there is a doctrine that in the comparison of hands:

• the highest-ranking should be that which is least likely to be dealt;
• the lowest-ranking should be that which is most likely to be dealt.
For several reasons, we have not adopted this principle, which often gives a result that differs from the two-step method of comparing hands:
• The two-step method is easy to remember, and readily extends to hands of fewer than or more than five cards, yielding a broader range of options for dealer's choice. Some multi-card statistics appear in table nine of the next section.
• The relative likelihoods of respective combinations depends on the presence or absence of wild cards. Because a player has some flexibility in designating how a wild card is to be used, there can be ambiguity in which categories are most likely to be dealt, and thus their relative ranking.
• The relative likelihoods of hands depends on how many ranks are in the pack and how many cards are in each rank. Below, an excerpt of table six reveals that v1111 occurs more frequently than v221 for packs with 13 or 12 ranks, but the situation is reversed with 10 or 8 ranks.

table six — excerpt
distribution by category
13 ranks12 ranks 10 ranks 8 ranks
v221 13.22%14.06% 15.53%16.04%
v1111 21.48%19.20% 14.72%10.60%

It would extremely cumbersome if players found it necessary to calculate the probabilities and remember which hands defeat which others for each of the many possible configurations. This is especially true if variants beyond those listed here are introduced; and Poker players frequently experiment with rule alterations. The two-step rule, applied under all circumstances, makes the game far more tractable.

Table nine shows the distribution of hands when other than five cards are dealt.

table nine
distribution by category
assuming 13 ranks and 4 cards per rank
three cards per hand  four cards per hand  five cards per hand
v3 704 3.19% v4 2,560 0.95% v5 9,216 0.35%
v21 7,616 34.46% v31 26,208 9.68% v41 89,088 3.43%
v111 13,780 62.35% v22 11,952 4.41% v32 76,896 2.96%
total 22,100100.00% v211 124,224 45.89% v311 399,984 15.39%
v1111 105,781 39.07% v221 343,584 13.22%
total 270,725100.00% v2111 1,121,952 43.17%
v11111 558,240 21.48%
total 2,598,960100.00%

six cards per hand  seven cards per hand  eight cards per hand
v6 32,768 0.16% v7 114,688 0.09% v8 393,216 0.05%
v51 298,496 1.47% v61 983,040 0.73% v71 3,170,304 0.42%
v42 246,656 1.21% v52 771,072 0.58% v62 2,334,720 0.31%
v411 1,268,608 6.23% v511 3,955,200 2.96% v611 12,091,392 1.61%
v33 117,064 0.58% v43 705,408 0.53% v53 2,059,392 0.27%
v321 2,037,120 10.01% v421 6,043,008 4.52% v521 17,451,264 2.32%
v3111 3,384,400 16.62% v4111 10,042,368 7.51% v5111 29,257,344 3.89%
v222 281,408 1.38% v331 2,835,024 2.12% v44 996,000 0.13%
v2211 4,057,824 19.93% v322 2,219,392 1.66% v431 15,621,888 2.08%
v21111 6,470,560 31.78% v3211 22,253,472 16.63% v422 6,004,224 0.80%
v111111 2,163,616 10.63% v31111 18,355,856 13.72% v4211 61,569,408 8.18%
total 20,358,520100.00% v2221 5,751,136 4.30% v41111 51,159,168 6.80%
v22111 27,120,960 20.27% v332 5,215,296 0.69%
v211111 26,162,752 19.56% v3311 28,633,572 3.80%
v1111111 6,471,184 4.84% v3221 41,601,728 5.53%
total 133,784,560100.00% v32111 138,210,816 18.37%
v311111 70,071,520 9.31%
v2222 2,612,700 0.35%
v22211 50,535,808 6.72%
v221111 118,998,120 15.81%
v2111111 79,063,424 10.51%
v11111111 15,486,846 2.06%
total 752,538,150100.00%

These vertical meld combinations can be used in almost any kind of Poker playing; some genres are listed below.

Players often speak of cards that are "face up" or "face down". The former applies to public cards that are literally lying face up on the table for all players to see. The latter refers to private cards that either are lying face down on the table, with the owner lifting a corner to peek; or are not touching the table at all but are instead held in the owner's hand so that only he can see them.

Straight Poker. Here, "straight" means "plain", and it does not refer to a sequence of cards. This is the simplest, and possibly original, Poker. Cards are dealt face down to the players, there is one betting interval, and then hands are shown.

Draw Poker. A representative format is this:

1. Cards are dealt face down to each player.
2. The first betting interval takes place.
3. Each player may discard some of his cards, obtaining replacements that are hoped to be better.
4. The second betting interval takes place.
5. Players show their cards, and best hand wins.

Stud Poker. A popular choice among the many versions has each player being dealt seven cards, selecting the five that will give him the best standard Poker hand, and ignoring the other two. However, with the vertical meld system of this report all seven cards can participate. In typical practice:

1. Each player is dealt a few cards, some face up and some face down.
2. A betting interval takes place.
3. Each player is dealt another card, either face up or down according to the rules of the variant.
4. Steps 2 and 3 are repeated until each player has a complete hand.
5. A final betting interval takes place.
6. Players show their cards, and best hand wins.

Community Card Poker. In this vast family of games, a few cards are dealt (usually face down) to each player, and a few cards are dealt (either face up or down) to the center of the table — these are the community cards. There are multiple betting intervals, and any face-down commmunity cards are turned up as the game progresses. At the end, each player forms the best five-card hand choosing from his own cards and the community cards. As with Stud, the vertical meld system allows all cards to figure into the final result, instead of just five.

"Mexican" Stud Poker. This follows the general principles of Stud, but all cards are dealt face down, and from time to time each player performs a rollover, turning one of his face-down cards face up. Of course, for rolling over a player tries to select whichever card will most confound his opponents, and this introduces great latitude for strategy. Rollovers can also be employed in Draw Poker.

With the vertical meld system, a player is often able to roll over several cards without giving his opponents any particular indication of how good his hand really is. The following is an example where each player was dealt seven cards, and at a later stage of the game has rolled over four of them. Suppose his face-up cards are Q-T-8-6, with three of his cards still face down. Then as far as his opponents can tell, he might hold almost any of the 15 seven-card categories of table nine, including the extremes v7 (QJT9876) and v1111111 (A-Q-Q-Q-T-8-6).

The two-step rule means that two Poker hands can be compared even if they have different numbers of cards: as always, whoever has the longest sequence wins.

Here is an illustration of how that works. Suppose in a modification of Draw Poker each player is dealt nine cards; and when drawing must discard two cards, receiving only one in return. After several rounds of betting and drawing, a player who has drawn four times now has five cards (v41 = 9876-2), while another who drew two times now has seven (v322 = KQJ-76-32). The player with the v41 wins because he has the longest sequence, even though his opponent has more cards.

When breaking ties under step one of the two-step rule, a player loses if he runs out of sequences before his opponent. For instance, 7654-2 defeats KQJT.