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
Rank		Symbol
highest	Ace	A
	King	K
	Queen	Q
	Jack	J
	Ten	T
	Nine	9
	Eight	8
	Seven	7
	Six	6
	Five	5
	Four	4
	Three	3
lowest	Two	2

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 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
category		description	examples
			highest	lowest	comparisons
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
category		description	example
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
category		description	example
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
category		description
highest	v5 and f5	straight flush
	h41	four of a kind
	f32	full house
	f5	flush
	v5	straight
	h311	three of a kind
	h221	two pair
	h2111	one pair
lowest	h11111	high 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	5,461,512	100.00%	2,598,960	100.00%	1,712,304	100.00%	658,008	100.00%	201,376	100.00%

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	575,757	100.00%	2,598,960	100.00%	8,259,888	100.00%	21,111,090	100.00%

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	1,712,304	100.00%	658,008	100.00%

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.

• 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 ranks	12 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	22,100	100.00%		v211	124,224	45.89%		v311	399,984	15.39%
				v1111	105,781	39.07%		v221	343,584	13.22%
				total	270,725	100.00%		v2111	1,121,952	43.17%
								v11111	558,240	21.48%
								total	2,598,960	100.00%
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	20,358,520	100.00%		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	133,784,560	100.00%		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	752,538,150	100.00%

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.