Leap sequences for card games.
Version of Thursday 28 February 2019.
Dave Barber's other pages.

§1. This report introduces a system of melds that could be used in a variety of Rummy games. The leap-sequence melds here are a generalized form of the usual sequences found in Rummy. On the other hand, nothing comparable to the traditional same-rank sets is proposed here.

The system will be described in terms of widely-used 13-rank 4-suit playing cards. Although the number of suits is not critical, an essential characteristic is that 13 is a prime number.

Every sequence of ranks is circular. This one is the fundamental:

… four – five – six – seven – eight – nine – ten – jack – queen – king – ace – two – three – four – five …

which can be abbreviated thus:

… 4 5 6 7 8 9 T J Q K A 2 3 4 5 …

For computational convenience, the jack can be regarded as an 11, queen 12, king 13, ace 1.

A sequence of more than 13 cards, containing repetition, can be achieved if multiple packs of cards are used, which is not uncommon in the Rummy family, as with Canasta. To use Rummy parlance, melds can go "around the corner". Some examples of sequences in this report display the repetition to make the circularity clearer.

Note that charts are numbered according to the sections of this report, so some numbers will be skipped.

§2. There are six kinds of sequence:

chart 2A:
thirteen ranks
leap-1… 4 5 6 7 8 9 T J Q K A 2 3 4 5 …
same as the fundamental
leap-2… 4 6 8 T Q A 3 5 7 9 J K 2 4 6 …
leap-3… 4 7 T K 3 6 9 Q 2 5 8 J A 4 7 …
leap-4… 4 8 Q 3 7 J 2 6 T A 5 9 K 4 8 …
leap-5… 4 9 A 6 J 3 8 K 5 T 2 7 Q 4 9 …
leap-6… 4 T 3 9 2 8 A 7 K 6 Q 5 J 4 T …

Because 6 + 7 = 13, a leap-7 sequence would be the same as a leap-6 in reverse, so the two are not distinguished; similarly for leap-8 and leap-5, et cetera. Also, there is no leap-0 or leap-13 sequence, which would contain cards all of the same rank.

The sequences become a little clearer if the Rook pack is used instead of the standard 52. It has four suits, symbolized by color; and 14 ranks, numbered 1 through 14 with no letters. To obtain 13 ranks, the 14s would be discarded, although some players might find them useful as wild cards.

Ideal is to have custom-printed cards numbered 0 through 12, making the modular arithmetic most conspicuous. If custom cards are going to be produced, a more elaborate approach with multiple ranks on each card, but no suits, might be chosen.

Adaptations for other numbers of ranks are discussed in §7 below.

§3. A valid meld is a sequence of at least 3 cards, all of the same suit, that are consecutive according to one of the leap sequences above. Melds are worth points; a suggested scoring schedule is the quadratic, which greatly rewards longer sequences.

chart 3A:
length of meldpoint valuerationale
3 11
4 31 + 2
5 61 + 2 + 3
6101 + 2 + 3 + 4
7151 + 2 + 3 + 4 + 5
8211 + 2 + 3 + 4 + 5 + 6
928similarly
1036
1145
1255
1366
n(n − 1) × (n − 2) ÷ 2
see triangular numbers

A 2-card meld is not recognized, in part because the traditional minimum in Rummy games is 3, and further because a 2-card meld, consisting of any two cards of the same suit but different rank, would be so easy to assemble that it does not deserve any points.

In chart 3B are some examples of melds. In every case, all cards within a meld are of the same suit. Neither the suit nor the leap affects the point value.

chart 3B:
examples of melds
meldleappoints
7-9-J 21
8-Q-3-7 43
A-7-K-6-Q 66
T-J-Q-K-A-2 110

As play continues, melds can be extended. For instance, 7-9-J (1 point) could grow to 5-7-9-J-K-2-4 (15 points).

The quadraticity has the effect of giving a bonus for large melds, but doing so on a sliding scale. Canasta, by comparison, gives only a small score for a six-card meld but abruptly awards a huge bonus when a seventh card is added.

Depending on the other rules of the game that the players choose, they may find it too difficult to make same-suit melds. One relaxation is to permit same-color melds, so that clubs and spades may be mixed, or hearts and diamonds. Going further is to ignore suits entirely.

§4. Although these leap-sequence melds can be employed in a great variety of card games, the rules for an example game are given beginning here and continued in §5.

Players: Two through six.

Equipment:

• One or more packs of 13-rank 4-suit playing cards.
• Six leaptokens, each bearing a number from one to six. These can be as simple as small pieces of cardboard with numbers written on them.
• A device for keeping score. Pencil and paper will work.

Object: To win by outscoring one's opponents.

Preliminaries:

• Players select seats.
• Players select a dealer, who shuffles. Some other player cuts.
• The dealer distributes five cards face down to each player. Players look at their cards secretly.
• Undealt cards are placed in a stack face down in the middle of the table, forming the stock.
• The leaptokens are placed near the middle of the table.

General procedure:

The player at dealer's left goes first. The turn will pass around the table clockwise.

Each player in turn has a choice between pursuing and retiring. A player who opts to pursue does this:

1. Draws two cards (if there are that many) from the top of of stock and adds them to their hand. If only one card remains, the player takes that. Even if the stock is empty, the player continues with steps 2 and 3.
2. Melds if willing and able, as described in §5 below.
3. Discards one card from their hand (if any remain) by placing it face down at the bottom of the stock. If a player has melded all their cards, no discard is required.

A player who opts to retire does this:

• Lays any remaining cards in their hand face down on the table, and announces a retirement.

A player who retires can never pursue again, but retains any points the they have earned. A player is permitted to retire for any reason. The game ends when all players have retired. Note that the game can continue with only one player if everyone else has retired.

At the end of the game, a player's score is the sum of the point values of their melds, minus the numerical value of any unmelded cards they retired with. (Ace = 1, Jack = 11, et cetera.) It is sufficient for players to not calculate their scores until the end of the game, but intense players will monitor everyone's scores as they change with each meld.

• In contrast to most Rummies, the game does not end when some player runs out of cards. Indeed, an unretired player with an empty hand can continue to pursue, drawing two cards (as available) from the stock and building up a hand again.

• In general, a player will not choose to retire until it becomes clear that they will be unable to do any further melding. The retirement rule exists for when an "infinite loop" of pointless plays is all that remain. Each player retires individually when that stage of pointlessness, in their opinion, has been reached.

• The draw-two-discard-one rule means that players' hands, in the absence of melding, will grow — whenever there are at least two cards in the stock. This explains the relatively small number (five) of cards dealt at the beginning of the game.

• The rule that a player discards face down at the bottom of the stock is quite unusual among the Rummies. The norm is that the discard goes face up at the top of a discard pile, but this game has no discard pile.

• Another unusual rule is that a player is excused from discarding at the end of their turn if they have no cards left.

§5. Here are the details of melding for the sample game of §4 above.

1. No player is ever required to meld.

2. To meld, a player lays several of their cards, all of the same suit, face up on the table clearly arranged for everyone to see. Each meld must conform to one of the leap sequences above. Each meld remains with its owner. No player may do anything to another player's meld.

3. No card can be a member of two melds. However, when multiple packs are used, the melding of one card does not restrict melding of the other card(s) equal to it.

4. There are two leap constraints:

• All of a player's melds must be of the same leap. For instance, one player cannot have both a leap-2 meld and a leap-3 meld.
• No two players can have melds of the same leap. For instance, as soon as one player melds a leap-4, no other player can meld a leap-4.

The leaptokens aid in enforcing this rule. When a player lays down their first meld, they take from the center of the table the leaptoken bearing the pertinent number. For instance, if their meld is leap-3, they take the #3 leaptoken. On the other hand, if the #3 leaptoken is already held by some other player, it would have been illegal for this player to have laid down a leap-3 meld.

5. On each turn, a player is permitted to do one (not both) of the following:

• Start a new meld that contains at least three cards.
• Extend one (and only one) of their previous melds with one or more cards at either or both ends.
Special note: If a player has two separate melds in the same suit, but when melding extends one so that it abuts the other, the player on that turn (not later) may conjoin them, and on that same turn add more cards to either end of the conjoined meld. Because of the leap constraints, it is guaranteed that both of the original melds must have been of the same leap. Such conjoining is desirable because it greatly increases a player's score. A leap-5 example begins with a player holding A-6-J (1 point) and 8-K-5-T (3 points). Then the player extends the second meld to 3-8-K-5-T (6 points). These two melds separately are worth 3 + 6 = 9 points. But conjoined, they become a 21-point meld: A-6-J-3-8-K-5-T.

6. Any legal meld is permanent, and cannot be retracted or modified, except to be extended or conjoined by its owner.

As a courtesy, the cards of a meld should be positioned in the same direction as the leap sequences given in chart 2A. For instance, it is better to arrange a meld as 4-9-A (leap 5) rather than A-9-4 (reverse leap 5).

Alternative to melding rule 4: All of a player's melds must be of different leaps, even two melds in the same suit. Hence melds can never be conjoined, as the result would not be of a single leap. Multiple players can, and often will, have melds of the same leap. Leaptokens are no longer needed.

§6. Some players will elect to use wild cards.

When the number of ranks is prime, a meld consisting of natural-wild-natural is never ambiguous; but the value for which the wild card stands may not be obvious, and this can lead to mistakes. Examples:

chart 6A:
thirteen ranks
3-card
meld
value of
wild card
leap
sequence
4-W-5J6 reverse
4-W-651
4-W-7Q5 reverse
4-W-862
4-W-9K4 reverse
4-W-T73
4-W-JA3 reverse
4-W-Q84
4-W-K22 reverse
4-W-A95
4-W-231 reverse
4-W-3T6
4-W-4illegal

For that reason, the following rule is recommended: Every meld must have at least one instance of two adjacent natural cards somewhere. This will make the value of the leap clear. Another rule that can help prevent mistakes is this: No meld can have two adjacent wild cards.

If the number of ranks is not prime, a natural-wild-natural meld can be ambiguous. Consider a pack with these 10 ranks:

… 4 5 6 7 8 9 T A 2 3 4 5 …

The meld 4-W-6 could validly be interpreted as 4-5-6 (step 1) or 4-T-6 (step 4 reverse), and players would need to invent a rule to address this. A direct solution is to require every meld to contain two adjacent natural cards, as suggested above.

§7. The number of ranks need not be 13. Prime numbers are viable candidates.

11 ranks. A pack of Spanish playing cards with 12 ranks can be adapted for leap sequences. The 12s should be discarded, leaving 11 ranks, 11 being a prime number. Modern Spanish cards are typically numbered, even the royalty cards.

There are five kinds of sequence:

chart 7A:
eleven ranks
leap-1… 4  5  6  7  8  9 10 11  1  2  3  4  5 …
leap-2… 4  6  8 10  1  3  5  7  9 11  2  4  6 …
leap-3… 4  7 10  2  5  8 11  3  6  9  1  4  7 …
leap-4… 4  8  1  5  9  2  6 10  3  7 11  4  8 …
leap-5… 4  9  3  8  2  7  1  6 11  5 10  4  9 …

The 12s might be used as wild cards, either without restriction, or each being limited to filling sequences in its own suit.

The five-suited proprietary game Five Crowns also uses 11 ranks: 3, 4, 5, 6, 7, 8, 9, T, J, Q, K. (The lack of aces and twos makes sense under the rules of that game.) Chart 7A can be adapted by substitution:

chart 7B:
eleven ranks (Five Crowns)
leap-1… 4 5 6 7 8 9 T J Q K 3 4 5 …
leap-2… 4 6 8 T Q 3 5 7 9 J K 4 6 …
leap-3… 4 7 T K 5 8 J 3 6 9 Q 4 7 …
leap-4… 4 8 Q 5 9 K 6 T 3 7 J 4 8 …
leap-5… 4 9 3 8 K 7 Q 6 J 5 T 4 9 …

7 ranks. A Spanish pack with 10 ranks can also be used. To obtain a prime number of ranks, the 8s, 9s, and 10s should be discarded. Now the game is limited to three players, unless multiple players are allowed to use the same leap.

chart 7C:
seven ranks
leap-1… 4 5 6 7 1 2 3 4 5 …
leap-2… 4 6 1 3 5 7 2 4 6 …
leap-3… 4 7 3 6 2 5 1 4 7 …

Four-suited Swiss and German packs of 36 or 32 cards could be similarly reduced to 7 ranks; the same applies to a Bezique or Skat pack.

5 ranks. A six-rank Pinochle pack (9 J Q K T A) can also be adapted by discarding the nines. Because a full Pinochle pack has 48 cards (6 ranks, 4 suits, 2 of each card), there will be 40 cards after removing the nines. Only two leaps are possible, so it will probably be necessary to allow multiple players to use the same leap. Pinochle traditionalists will want to position the ten between the king and ace.

chart 7D:
five ranks (Pinochle)
leap-1… J Q K T A J Q …
leap-2… J K A Q T J K …

17 ranks. This elaboration, which requires custom-made cards, makes 8 different leaps possible. The following example assumes the cards to be numbered 0 through 16:

chart 7E:
seventeen ranks
leap-1… 4  5  6  7  8  9 10 11 12 13 14 15 16  0  1  2  3  4  5 …
leap-2… 4  6  8 10 12 14 16  1  3  5  7  9 11 13 15  0  2  4  6 …
leap-3… 4  7 10 13 16  2  5  8 11 14  0  3  6  9 12 15  1  4  7 …
leap-4… 4  8 12 16  3  7 11 15  2  6 10 14  1  5  9 13  0  4  8 …
leap-5… 4  9 14  2  7 12  0  5 10 15  3  8 13  1  6 11 16  4  9 …
leap-6… 4 10 16  5 11  0  6 12  1  7 13  2  8 14  3  9 15  4 10 …
leap-7… 4 11  1  8 15  5 12  2  9 16  6 13  3 10  0  7 14  4 11 …
leap-8… 4 12  3 11  2 10  1  9  0  8 16  7 15  6 14  5 13  4 12 …

Playing card manufacturers often print their product in sheets of 56 cards: 52 rank-and-suit, 2 Jokers, 2 miscellaneous. A pack of 17 ranks and 3 suits (51 cards) could be conveniently produced with current technology, and there would still be room for wild cards.

Non-prime ranks. In this case, there can be multiple sequences, out of phase with one another, for any given leap, as enumerated below in the case of 12 ranks. The game becomes more complicated. With a single pack, some sequences might be no longer than two cards, and thus not meldable in the absence of a special rule.

chart 7F:
twelve ranks
leap-1 … 4 5 6 7 8 9 T J Q A 2 3 4 5 …
leap-2 … 4 6 8 T Q 2 4 6 …
… 5 7 9 J A 3 5 7 …
leap-3 … 4 7 T A 4 7 …
… 5 8 J 2 5 8 …
… 6 9 Q 3 6 9 …
leap-4 … 4 8 Q 4 8 …
… 5 9 A 5 9 …
… 6 T 2 6 T …
… 7 J 3 7 J …
leap-5 … 4 9 2 7 Q 5 T 3 8 A 6 J 4 9 …
leap-6 … 4 T 4 T …
… 5 J 5 J …
… 6 Q 6 Q …
… 7 A 7 A …
… 8 2 8 2 …
… 9 3 9 3 …