Contents:
§1 Introduction. The Game of Trains (GoT) is a game, combining chance and skill, for several players using custom playing cards. The object is for each player to transform their sequence of numbered playing cards from descending order into ascending; the numbers need not be consecutive. For example, the sequence 3162248507880 qualifies. The first player to do this wins.
This web page introduces a family of games resembling Game of Trains, but with modified rules, and requiring differentlyprinted cards.
As supplied by the factory, the Game of Trains has 88 cards displaying caricatures of railroad rolling stock and cargo:
The railroad theme is only decorative, and the game can be played as an abstract numerical endeavor. The sample playing cards illustrated on this page are as plain as possible, omitting the standard GoT artwork which is covered by copyright.
The quantity of cards in each sequence is termed its census. Important is that all players are working on sequences of the same census. The factory's rules call for a census of seven, but other numbers are certainly possible, and a smaller census may be preferred when small children are playing.
A virtue of GoT and these variants is that if the pack becomes defective through the loss of a few cards, the game is not spoiled. Of course, serious players will want to know exactly which cards are missing.
While GoT is clearly a relative of RackO, there are significant differences. Greatly affecting strategy is that the values of a GoT player's cards can be seen by opponents, but a RackO player's cannot. Also, a player's original sequence in GoT is necessarily descending, while in RackO it is random; in this regard, GoT reduces the level of chance compared to RackO.
§2 Placemats. Each player in these variants should have a placemat, which can be as a simple as a piece of heavy paper printed as in figure 1. For a censusseven game, the placemat will have seven spaces to hold the player's seven cards. Each space has a letter for reference:
— figure 1 — 

During play, each space will contain precisely one card, although one card frequently replaces another as the game progresses. With the placemat in use, the locomotive cards provided in the factory pack are no longer needed; they might find service as wild cards.
§3 Pattern one. Described first is the censusseven version of pattern one.
There are 21 possible twocard swaps (for instance A ↔ B, D ↔ F, E ↔ G) of two cards in the same sequence. These cards support all of them, but no other abilities. On each card are printed its number and two letters representing its swapping ability:
— figure 2 —  

The characters in small type are for the convenience of a player holding several cards in their hand, allowing them to arrange the cards into a compact fan, as players might prefer if playing by nonfactory rules. Meanwhile, the characters in large type make it easy to identify a faceup card lying on the table from a distance. Too, these cards are doubleended (unlike the factory product), so they look the same if rotated 180 degrees, as might happen in ordinary play.
Countless typographical and artistic variations are possible in the design of the cards. The realizations here are close to minimal, employing the Scalable Vector Graphics generic font family monospace, which should yield decent results on all browsers.
Pattern one includes all the twocard swaps of the factory pack, and many more. Not supported are threecard rotations, removals, or protection.
Figure 3 shows a placemat loaded with seven cards in descending numerical order, as at the beginning of a game:
— figure 3 —  

— the same, condensed —  

If this player is the first to get seven cards (not necessarily the original seven) in ascending numerical order, this player wins.
The swaps and numbers go together as follows:
pattern one, census seven  

ability  cards  ability  cards  ability  cards  
A ↔ B  1, 22, 43, 64  B ↔ D  8, 29, 50, 71  C ↔ G  15, 36, 57, 78  
A ↔ C  2, 23, 44, 65  B ↔ E  9, 30, 51, 72  D ↔ E  16, 37, 58, 79  
A ↔ D  3, 24, 45, 66  B ↔ F  10, 31, 52, 73  D ↔ F  17, 38, 59, 80  
A ↔ E  4, 25, 46, 67  B ↔ G  11, 32, 53, 74  D ↔ G  18, 39, 60, 81  
A ↔ F  5, 26, 47, 68  C ↔ D  12, 33, 54, 75  E ↔ F  19, 40, 61, 82  
A ↔ G  6, 27, 48, 69  C ↔ E  13, 34, 55, 76  E ↔ G  20, 41, 62, 83  
B ↔ C  7, 28, 49, 70  C ↔ F  14, 35, 56, 77  F ↔ G  21, 42, 63, 84 
Each swap appears once in each quadrant (121; 2242; 4363; 6484) of the pack. A twoplayer game might benefit from a smaller pack, using cards 142; three players, 163. Within either of these reductions, no swap is present more times than any other. Too, there is a obvious way to extend the pack to 105 or 126 cards if more than four are playing.
A censussix game can be played by removing all the cards that specify space G in a swap (6, 11, 15, 18, 20, 21, et cetera). It is better, however, to obtain a sixspace placemat and to use the following 60card pack:
pattern one, census six  

ability  cards  ability  cards  ability  cards  
A ↔ B  1, 16, 31, 46  B ↔ C  6, 21, 36, 51  C ↔ E  11, 26, 41, 56  
A ↔ C  2, 17, 32, 47  B ↔ D  7, 22, 37, 52  C ↔ F  12, 27, 42, 57  
A ↔ D  3, 18, 33, 48  B ↔ E  8, 23, 38, 53  D ↔ E  13, 28, 43, 58  
A ↔ E  4, 19, 34, 49  B ↔ F  9, 24, 39, 54  D ↔ F  14, 29, 44, 59  
A ↔ F  5, 20, 35, 50  C ↔ D  10, 25, 40, 55  E ↔ F  15, 30, 45, 60 
Extensions to larger and smaller censuses can be handled similarly.
§4 Pattern two. This has 28 abilities, including all the swaps of pattern one, and adding seven onecard removals, similar to the removals of the factory pack. A symbol like A → X means to remove the card at space A of the sequence and replace it with a new card from the draw pile.
— figure 4 —  

pattern two, census seven  

ability  cards  ability  cards  ability  cards  ability  cards  
A → X  1, 29, 57  B → X  8, 36, 64  C ↔ D  15, 43, 71  D ↔ G  22, 50, 78  
A ↔ B  2, 30, 58  B ↔ C  9, 37, 65  C ↔ E  16, 44, 72  E → X  23, 51, 79  
A ↔ C  3, 31, 59  B ↔ D  10, 38, 66  C ↔ F  17, 45, 73  E ↔ F  24, 52, 80  
A ↔ D  4, 32, 60  B ↔ F  11, 39, 67  C ↔ G  18, 46, 74  E ↔ G  25, 53, 81  
A ↔ E  5, 33, 61  B ↔ G  12, 40, 68  D → X  19, 47, 75  F → X  26, 54, 82  
A ↔ F  6, 34, 62  B ↔ H  13, 41, 69  D ↔ E  20, 48, 76  F ↔ G  27, 55, 83  
A ↔ G  7, 35, 63  C → X  14, 42, 70  D ↔ F  21, 49, 77  G → X  28, 56, 84 
§5 Application of abilities. With either of these patterns, the players can simply follow the factory rules for the game. There is, however, an alternative that gives players more room for judgement: The player who uses a card for its ability has the option of applying it either to their own sequence, or to the sequence of an opponent of their choice.
In a game with 84 cards and 7 spaces, it will often be true of a completed sequence that:
To give examples of using the cards' abilities, suppose at some point in a patterntwocensusseven game four players' sequences look like this:
— figure 5 —  

 
 
 

57 AX  If Carl draws card 57AX, he could profitably apply it to his own sequence, because he has a large number at space A and would likely draw a smaller number as replacement. Carl could also benefit by applying it to Andy, who has a small number at space A and would probably draw a larger number as replacement. 
10 BD  If Beth draws 10BD, she could apply it to herself, swapping cards 37 and 12 and bringing her sequence closer to fully ascending. She should not apply it to Carl, or he would enjoy a similar benefit. Applied to Andy or Dora, it would make little difference. 
There are mathematical ways to express the "disorder" of a sequence; no one way is ideal for all purposes. One popular way is to count the number of inversions. In the case of Dora, there are 8 (out of a maximum of 21):
A different way to evaluate disorder is to find the minimum number of swaps necessary to put a sequence in order. Here is an extreme example:
— figure 6 —  


Although the sequence is descending, three suitablychosen swaps are enough to make it ascending: A ↔ G, B ↔ F, C ↔ E. Of course, the player must assess the likelihood of obtaining cards with those three abilities on a timely basis.
§6 Pattern three. In patterns one and two, any swapping ability was limited to two cards in the same sequence. However, that restriction is not inevitable.
To illustrate an extension of the rules, consider an example with players Earl, Fran, and Gary. If Earl wants to apply his card with the B ↔ D swapping ability, and if he chooses to swap with Fran, then Earl could either swap his Bspace card with her Dspace card; or his Dspace card with her Bspace card.
Some rulemakers may go further, allowing Earl to direct Fran to swap her Bspace card with Gary's Dspace card, or vice versa.
A remaining rule question is whether players will still be allowed to swap two cards within the same sequence.
This flexibility in swapping greatly increases the number of possible swaps, and consequently the amount of thinking time a player requires to make a wise choice; this consideration becomes more critical as the number of players increases. For that reason, it is suggested that the census be reduced, to perhaps five, so as not to prolong the game.
Here is what a pack might look like:
pattern three, census five  

ability  cards  ability  cards  
A ↔ B  1, 11, 21, 31, 41  B ↔ D  6, 16, 26, 36, 46  
A ↔ C  2, 12, 22, 32, 42  B ↔ E  7, 17, 27, 37, 47  
A ↔ D  3, 13, 23, 33, 43  C ↔ D  8, 18, 28, 38, 48  
A ↔ E  4, 14, 24, 34, 44  C ↔ E  9, 19, 29, 39, 49  
B ↔ C  5, 15, 25, 35, 45  D ↔ E  10, 20, 30, 40, 50 
§7 Multiple packs. When two packs are combined, numbers will be duplicated. In establishing rules, players need to appreciate that "descending sequence" and "neverascending sequence" are not quite synonyms; similarly contrasted are "ascending sequence" and "neverdescending sequence". Definitions:
ascending  each card is less than every card on its right 

neverdescending  each card is less than or equal to every card on its right 
descending  each card is greater than every card on its right 
neverascending  each card is greater than or equal to every card on its right 
If a sequence has repeated numbers, it does not qualify as descending or ascending, although it might still be neverascending or neverdescending respectively. Still, descending is assuredly neverascending; and ascending is certainly neverdescending. (see monotonic) Examples:
descending  never ascending  sequence  never descending  ascending 

yes  yes  7654321  no  no 
no  yes  7644321  
no  no  1234467  yes  no 
1234567  yes  yes 
If so many packs are used that all the cards in some sequence are equal, it is both neverascending and neverdescending.
The key game design question is whether repeated numbers should be allowed in a winning sequence.
§8 Two dimensions. There is a twodimensional Game of Thrones variation of census nine, using three rows and three columns. Figure 6 shows the placemat, where each space is identified by two letters, such as AD for the top left, BF for the center right, and so forth.
— figure 7 — 

At the beginning of the game, nine cards are randomly dealt to each player, who arranges them in any way such that each of the rows, and each of the columns, is in descending order. There are 42 ways to do this.
Using the factory's rules or some variant, the first player to achieve ascending order in all rows and all columns wins.
The twodimensionality of the arrangement of players' cards provides a wide variety of plausible abilities. Below are tables with 16 categories of suggestions; game designers will probably elect to support only a few. Contrast that some of these proposed abilities are specific, leaving no player with any choice in how to carry them out; others are discretionary, permitting some flexibility in their use.
The rationale for having discretionary abilities is to give a player a decent opportunity to improve their own row or column without damaging the intersecting columns or rows; or to foul an opponent's row or column without improving the intersecting colums or rows. The player who applies an ability to an opponent is the one who gets to do the choosing — the opponent is a passive victim.
It is recommended that any ability that pertains to rows have a counterpart that pertains to columns.
The following are six categories of specific abilities that might be supported:
Swapping two specific cards within the same row (9 ways):
 Swapping two specific cards within the same column (9 ways):
 
Swapping two specific cards that share neither a row nor a column (18 ways):
 
Rotating within a specific row (6 ways):
 Rotating within a specific column (6 ways):
 
Removing (and then replacing) a specific card (9 ways):

Next are ten categories of discretionary abilities. The difference between some of them can be subtle:
Swapping the player's choice of:
 Swapping the player's choice of:

Swapping within the player's choice of rows:
 Swapping within the player's choice of columns:

Horizontal rotations:
 Vertical rotations:

Horizontal rotations:
 Vertical rotations:

Removals (with replacement):
 Removals (with replacement):

No doubt, someone will extend this into a threedimensional version.
§9 Numberbased abilities. All the abilities described thus far have been based on the positions of the cards on the placemat. There is, however, a range of abilities relying on the numbers printed on the cards. Although the matter will not be fully developed here, some examples should be mentioned:
An ability like "remove the lowest evennumbered card" will fail if all of a player's cards are oddnumbered. Game designers might avoid including such contingent abilities entirely. Alternatively, they might establish a rule that a player may not attempt to apply an impossible ability.
§10 Wild cards. It is possible to play the Game of Trains with wild cards.
One kind of wild card would represent any number its owner chooses, perhaps changing as the game progresses, and would lack any abilities. Most players would be willing to recognize the following as a valid ascending sequence:
261438W5682
but they might reject these:
261438W3982  (W = 38½) 
2614383984W  (W = 85) 
because the wild card would have to stand for some number that does not appear on any card in the pack.
A sequence with a wild card is dirty; without a wild card it is clean.
With at least three players, game points might be awarded as follows:
A player who completes a dirty sequence has a strategic choice to make:
§11 Freezing. Each player needs several small tokens such as coins, buttons, or checkers. They need not be of uniform design.
At the beginning of a player's turn, they can freeze one card of their sequence by placing a token on it. After that, this card can never be taken from this space by swapping, replacement, removal, et cetera; by the owner or any other player. A frozen card can never be unfrozen, which differs from the factory's rules for use of a protection card.
After freezing a card, the player takes a normal turn.
Freezing is most attractive for a card of very low number in space A, or very high number in space G (assuming a censusseven game). Still, any card in the sequence is permitted to be frozen.