§1. This report describes a system of playing cards, dubbed LINMODSEQ, intended for forming sequences.
Each card bears several ordinary digits, requiring no other symbols; as a result, the design of the cards can be very simple. Moreover, no particular size or shape is required, and they can be printed in one color using ordinary typography. For convenience, the same digit combination is printed in four places on the card, in two sizes. Examples:
The number of digits in the combination is the card's multiplicity. In figure one, the multiplicities are 1, 2, 3, and 4 respectively. Presumably, all cards in a pack would be of the same multiplicity.
The digits in small type ("squeezers") are for a player holding several cards in their hand, allowing them to arrange the cards into a compact fan. The digits in large type make it easy to identify a face-up card from a distance, and are underscored to reduce confusion between a 6 and a 9.
§2A. What follows is a detailed example of a pack described as modulus 7 (explained below) and multiplicity 2 ("mod-7-mult-2"). These values were selected for detailed discussion because they produce a pack large enough to be interesting but small enough to be convenient. Each of the digits is between 0 and 6 inclusive; this a key characteristic of modulus 7. Each of the 49 = 7 × 7 possible cards appears once within the pack. Examples:
Two cards having the same digits in a different order, as 26 and 62, are not the same thing. In that aspect, these cards differ from dominoes.
§2B. Use of these LINMODSEQ cards is based on a linear modular sequence, "LMS" for short. (This is the same as the modular linear sequence of the MODLINSEQ board game, but the adjectives in the name have been switched to reduce confusion between the games.) Here is an example of an LMS, with hyphens (not minus signs) between the digits:
… 5-6-0-1-2-3-4-5-6-0-1-2 …
Each digit is 1 greater than the digit to its left, but if any digit would be greater than 6, then 7 is subtracted from it. From another point of view, each digit is 1 less than the digit to its right, but if any digit would be less than 0, then 7 is added to it. This reflects the fundamentals of how modulus 7 works. The next two examples have increments of 2 and 3 respectively:
… 3-5-0-2-4-6-1-3-5-0-2-4 …
… 1-4-0-3-6-2-5-1-4-0-3-6 …
An increment of 4 produces the reverse of the sequence that has increment 3, and in LINMODSEQ games the two would (presumably) regarded as the same; similarly with 5 and 2, or 6 and 1. An increment of 0 might also occur, as with:
… 2-2-2-2-2-2-2-2-2-2-2-2 …
For more, see modular arithmetic.
§2C. Any two cards can be deemed consecutive members of a sequence, which can then be extended on either end. For instance, 60-32 can be preceded by 25 and followed by 04.
The first digits on the cards form one LMS, and the second digits form another. The two LMSs are independent, and might have different increments.
Extended to the point of cycling around, 60-32 becomes:
Note that the sequence of first digits is 5-2-6-3-0-4-1 and the sequence of second digits is 3-5-0-2-4-6-1, each an ordinary LMS in its own right.
Since there are no duplicate cards in this pack, no sequence can be longer than seven cards.
§2D. There are 168 possible sequences with the mod-7-mult-2 pack. Fortunately, there is no need for a player to try to memorize them, because they can easily be generated by the linear modular sequence rule. However, the listing may aid the novice in grasping the range of possibilities.
§3. Helpful in notation is lexicographic order. Of any two cards, the one that has the smaller first digit is said to precede the other; if the respective first digits match, the respective second digits are compared. For instance, 26 precedes 62, and 14 precedes 15.
Although it is correct to write the following sequence thus:
the canonical way is this:
Explanation: 04, being first in lexicographic order among the seven cards, appears first. Of the two cards adjacent to 04, 32 precedes 46 lexigraphically, so 32 is written second in the sequence, leaving 46 to appear last. A sequence will often need to be reversed for canonicalization.
§4. Analogous to the mod-7-mult-2 pack is the more compact mod-5-mult-2 pack, with 25 = 5 × 5 cards. It has 60 possible sequences. Some players might prefer to use two instances of each card, making a 50-card pack, enabling a sequence of length 10, such as:
A much larger pack is mod-11-mult-2, with 121 = 11 × 11 cards. It has 660 possible sequences. For typographical simplicity, the letter A is used for the number 10:
With even greater moduli would be introduced the letters B = 11, C = 12, et cetera, as is done in hexadecimal arithmetic.
Another option is the mod-5-mult-3 pack (not illustrated) which has 125 = 5 × 5 × 5 cards, all different. There are 1550 possible sequences, all of length 5, with examples here. The mod-3-mult-4 pack could be used for the Set Game.
§5A. When the modulus is a prime number, such as 5, 7, or 11, the possible sequences achievable with a particular pack are all the same length. For a composite (i.e. non-prime) number, however, things become more complicated, a fact will likely appeal to some players but not others. Sufficient to illustrate what can happen is the mod-6-mult-2 pack, with 36 = 6 × 6 cards.
Some LMSs are of period 6, incorporating all six values, as:
… 4-5-0-1-2-3-4-5-0-1-2 … (inc 1)
By contrast, some LMSs are of period 3, each using only one-half of the values:
… 4-0-2-4-0-2-4-0-2 … (inc 2)
… 5-1-3-5-1-3-5-1-3 … (inc 2)
Further, some LMSs are of period 2, each using only one-third of the values:
… 3-0-3-0-3-0-3 … (inc 3)
… 4-1-4-1-4-1-4 … (inc 3)
… 5-2-5-2-5-2-5 … (inc 3)
Finally, a constant LMS can be regarded as of period 1:
… 4-4-4-4-4-4-4 … (inc 0)
§5B. There are 174 possible sequences possible with this pack. They are of length 2, 3, or 6. In the table below are examples, including the quantity of each variety of sequence:
|period of second digit|
The case where both periods equal 1 would be trivially satisfied by any single card. Unclear, however, is whether any card games would find it beneficial to recognize a one-card sequence.
If for instance a triple pack is employed, then a sequence such as 43-43-43 could be built. When duplicates are present, the quantities in the table above would of course have to be greatly increased.
§5C. If two cards are chosen at random from the 36-card mod-6-mult-2 pack described above, there is a 3-in-35 chance they will define a 2-card sequence; 8-in-35 for a 3-card; and 24-in-35 for a 6-card.
If the shorter sequences make it hard to construct desirable rules for a game being invented, players might agree to prohibit them entirely. At this stage, it will no longer be true that any two cards define a legal sequence. With the mod-6-mult-2 pack, and with most other moduli and multiplicities, there will still be plenty of full-length sequences available to keep play interesting.
§6A. Some players enjoy using packs to which wild cards have been added. With LINMODSEQ cards, it is possible for one of the positions to be wild (symbolized by X) while the others are not. When the card is used, the X will represent whatever digit is required to satisfy the sequence. Examples:
A card with more Xs is said to be wilder than a card with fewer Xs. With multiplicity 3 for instance, XXX is wilder than X2X, which itself is wilder than 32X, which in turn is wilder than 326.
Of course, players will have various opinions on what wild cards a pack ought to include. A gamut of wild cards for a mod-7-mult-2 pack contains 15 cards:
0X 1X 2X 3X 4X 5X 6X
X0 X1 X2 X3 X4 X5 X6
These would bring the total size of a mod-7-mult-2 pack to 49 + 15 = 64, some 23% of which are wild. Games with a large proportion of wild cards are often disfavored by expert players, as they are felt to excessively displace the need for skill, replacing it with chance. Note however that 14 of these cards are merely half-wild-half-natural, greatly reducing the extent to which they bring luck into the game.
§6B. In forming a sequence that includes wild cards, there is risk of ambiguity if nowhere in the sequence are two natural cards that are adjacent to each other. Here is an example of that from a mod-8-mult-2 pack:
00-X3-26 could stand for either of these:
Staying with a mod-8-mult-2 pack, this next sequence, with two adjacent natural cards and many wild cards, is unambiguous:
It means this:
In order to reduce mistakes in interpretation, players might opt to limit how many wild cards can be used in a sequence, or to stipulate that two wild cards can never be adjacent in a sequence.
When wild cards are used, the length of a sequence might exceed the modulus. In the example below, of length 9 with modulus 8, the last wild card equals the first card:
§6C. A sequence without any adjacent natural cards might still be unambiguous. For instance, under modulus 7 the cards 00-XX-33 cannot stand for anything except 00-55-33. Determining the necessary and sufficient conditions for non-ambiguity is a subtle matter.
Players might adopt a rule from one of these two categories:
|A.||Any ambiguity must be resolved when the sequence is established:|
|A1.||Every sequence must somewhere have two natual cards in adjacent positions, assuredly preventing ambiguity.|
|A2.||A sequence need not have two adjacent natural cards if the owner can somehow demonstrate non-ambiguity.|
|A3.||The owner of an otherwise-ambiguous sequence must announce the values of the wild cards, which cannot later be changed.|
|B.||Ambiguity need not be resolved when the sequence is established, possibly being resolved later when a card is added or replaced:|
|B1.||A previously-established ambiguous sequence may be extended with wild or natural cards.|
|B2.||Within a previously-established ambiguous sequence, a player may replace a card of greater wildness with a card of lesser wildness, if all natural digits are unchanged.|
|All new cards must be jointly consistent with at least one legal interpretation of the sequence as it stood before the change took place.|
When a wild card is placed in a sequence, there might be a partial reduction of ambiguity. In 00-XX-26, both the first and second sequences are ambiguous. Either of the extentions 00-XX-26-3X or 00-XX-26-X1 resolves one of the two ambiguities, as does either of the substitutions 00-1X-26 or 00-X3-26.
When the multiplicity is at least two, it is possible for the first digits to be of one modulus, and the second digits of another. Example with a pack of multiplicity 2, and moduli 5 and 7. To aid players in remembering that the various moduli are, it may be advisable to indicate on each card the modulus of each digit position.
Players might combine cards of different multiplicities, for example combining mod-4-mult-3, mod-4-mult-2, and mod-4-mult-1 packs for a total of 64 + 16 + 4 = 84 cards. Any sequence would be be limited to cards of only one multiplicity.
A mod-10-mult-2 pack can be reinterpreted as a mod-100-mult-1 pack. Then large increments become possible:
17-60-03-46-89-32-75-18-61 (mod 100, inc 43)
In the sequence above, the second digits happen to be an LMS with increment 3, because the modulus is a multiple of 10. However, the first digits are not an LMS because their increment is sometimes 4 and sometimes 5. The variation is due to carrying from the second digit position.
Higher-numbered cards can be eliminated if a smaller pack is desired. Here is a sequence using cards 00 through 62, rejecting 63 through 99:
18-32-46-60-11-25-39-53-04 (mod 63, inc 14)
In this second sequence, neither the first digits nor the second digits form an LMS.
XX would probably be the only wild card. As the two digits are construed as one number, a form like X3 or 7X is of uneven usefulness.