Game of Trains variants.
Version of Thursday 14 March 2019.
Dave Barber's other pages.

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 3-16-22-48-50-78-80 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 differently-printed cards.

As supplied by the factory, the Game of Trains has 88 cards displaying caricatures of railroad rolling stock and cargo:

• 4 locomotives; their only purpose is to remind players of where the beginning of their sequence is.
• 84 non-locomotives, each bearing a number from 1 to 84, with an indicator of an ability to manipulate one or more players' sequences.

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 Rack-O, there are significant differences. Greatly affecting strategy is that the values of a GoT player's cards can be seen by opponents, but a Rack-O player's cannot. Also, a player's original sequence in GoT is necessarily descending, while in Rack-O it is random; in this regard, GoT reduces the level of chance compared to Rack-O.

§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 census-seven 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 census-seven version of pattern one.

There are 21 possible two-card 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 non-factory rules. Meanwhile, the characters in large type make it easy to identify a face-up card lying on the table from a distance. Too, these cards are double-ended (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 two-card swaps of the factory pack, and many more. Not supported are three-card 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 —
 A B C D E F G 76CE 70BC 58DE 51BE 43AB 22AB 9BE

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
abilitycards  abilitycards  abilitycards
A ↔ B 1, 22, 43, 64 B ↔ D 8, 29, 50, 71 C ↔ G15, 36, 57, 78
A ↔ C 2, 23, 44, 65 B ↔ E 9, 30, 51, 72 D ↔ E16, 37, 58, 79
A ↔ D 3, 24, 45, 66 B ↔ F10, 31, 52, 73 D ↔ F17, 38, 59, 80
A ↔ E 4, 25, 46, 67 B ↔ G11, 32, 53, 74 D ↔ G18, 39, 60, 81
A ↔ F 5, 26, 47, 68 C ↔ D12, 33, 54, 75 E ↔ F19, 40, 61, 82
A ↔ G 6, 27, 48, 69 C ↔ E13, 34, 55, 76 E ↔ G20, 41, 62, 83
B ↔ C 7, 28, 49, 70 C ↔ F14, 35, 56, 77 F ↔ G21, 42, 63, 84

Each swap appears once in each quadrant (1-21; 22-42; 43-63; 64-84) of the pack. A two-player game might benefit from a smaller pack, using cards 1-42; three players, 1-63. 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 census-six 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 six-space placemat and to use the following 60-card pack:

pattern one, census six
abilitycards  abilitycards  abilitycards
A ↔ B 1, 16, 31, 46 B ↔ C 6, 21, 36, 51 C ↔ E11, 26, 41, 56
A ↔ C 2, 17, 32, 47 B ↔ D 7, 22, 37, 52 C ↔ F12, 27, 42, 57
A ↔ D 3, 18, 33, 48 B ↔ E 8, 23, 38, 53 D ↔ E13, 28, 43, 58
A ↔ E 4, 19, 34, 49 B ↔ F 9, 24, 39, 54 D ↔ F14, 29, 44, 59
A ↔ F 5, 20, 35, 50 C ↔ D10, 25, 40, 55 E ↔ F15, 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 one-card 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
abilitycards  abilitycards  abilitycards  abilitycards
A → X 1, 29, 57 B → X 8, 36, 64 C ↔ D15, 43, 71 D ↔ G22, 50, 78
A ↔ B 2, 30, 58 B ↔ C 9, 37, 65 C ↔ E16, 44, 72 E → X23, 51, 79
A ↔ C 3, 31, 59 B ↔ D10, 38, 66 C ↔ F17, 45, 73 E ↔ F24, 52, 80
A ↔ D 4, 32, 60 B ↔ F11, 39, 67 C ↔ G18, 46, 74 E ↔ G25, 53, 81
A ↔ E 5, 33, 61 B ↔ G12, 40, 68 D → X19, 47, 75 F → X26, 54, 82
A ↔ F 6, 34, 62 B ↔ H13, 41, 69 D ↔ E20, 48, 76 F ↔ G27, 55, 83
A ↔ G 7, 35, 63 C → X14, 42, 70 D ↔ F21, 49, 77 G → X28, 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:

• the card in space A has a number in or near the range 1-12,
• B: in or near 13-24,
• C: 25-36,
• D: 37-48,
• E: 49-60,
• F: 61-72,
• G: 73-84.
This information can be used to help a player judge which cards in their sequence are most severely out of order, and thus need the greatest attention. These ranges can be adjusted for other numbers of cards or spaces.

To give examples of using the cards' abilities, suppose at some point in a pattern-two--census-seven game four players' sequences look like this:

— figure 5 —
 Andy A B C D E F G 8BX 64BX 72CE 62AF 24EF 3AC 17CF
 Beth A B C D E F G 46CG 37BC 5AE 12BF 42CX 47DX 40BF
 Carl A B C D E F G 69BG 63AG 20DE 31AC 68BF 70CX 34AF
 Dora A B C D E F G 53EG 7AG 45CF 9BC 50DG 27FG 56GX

 57AX 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. 10BD 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):

• 53 > 7
• 53 > 45
• 53 > 9
• 53 > 50
• 53 > 27
• 45 > 9
• 45 > 27
• 50 > 27
When the number of inversions is zero, the sequence is fully ascending.

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 —
 A B C D E F G 70CX 69BG 68BF 63AG 34AF 31AC 20DE

Although the sequence is descending, three suitably-chosen 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 B-space card with her D-space card; or his D-space card with her B-space card.

Some rulemakers may go further, allowing Earl to direct Fran to swap her B-space card with Gary's D-space 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
abilitycards  abilitycards
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 ↔ E10, 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 "never-ascending sequence" are not quite synonyms; similarly contrasted are "ascending sequence" and "never-descending sequence". Definitions:

ascending each card is less than every card on its right each card is less than or equal to every card on its right each card is greater than every card on its right 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 never-ascending or never-descending respectively. Still, descending is assuredly never-ascending; and ascending is certainly never-descending. (see monotonic) Examples:

descendingnever-
ascending
sequencenever-
descending
ascending
yes yes 7-6-5-4-3-2-1 no no
no yes 7-6-4-4-3-2-1
no no 1-2-3-4-4-6-7 yes no
1-2-3-4-5-6-7 yes yes

If so many packs are used that all the cards in some sequence are equal, it is both never-ascending and never-descending.

The key game design question is whether repeated numbers should be allowed in a winning sequence.

§8 Two dimensions. There is a two-dimensional 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 two-dimensionality 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):

 AD ↔ AE AE ↔ AF AF ↔ AD BD ↔ BE BE ↔ BF BF ↔ BD CD ↔ CE CE ↔ CF CF ↔ CD

Swapping two specific cards within the same column (9 ways):

 AD ↔ BD BD ↔ CD CD ↔ AD AE ↔ BE BE ↔ CE CE ↔ AE AF ↔ BF BF ↔ CF CF ↔ AF

Swapping two specific cards that share neither a row nor a column (18 ways):

 AD ↔ BE AD ↔ BF AE ↔ BF AE ↔ BD AF ↔ BD AF ↔ BE BD ↔ CE BD ↔ CF BE ↔ CF BE ↔ CD BF ↔ CD BF ↔ CE CD ↔ AE CD ↔ AF CE ↔ AF CE ↔ AD CF ↔ AD CF ↔ AE

Rotating within a specific row (6 ways):

 rotatingright: AD → AE → AF → AD BD → BE → BF → BD CD → CE → CF → CD rotatingleft: AF → AE → AD → AF BF → BE → BD → BF CF → CE → CD → CF

Rotating within a specific column (6 ways):

 rotatingdown: AD → BD → CD → AD AE → BE → CE → AE AF → BF → CF → AF rotatingup: CD → BD → AD → CD CE → BE → AE → CE CF → BF → AF → CF

Removing (and then replacing) a specific card (9 ways):

 AD → X AE → X AF → X BD → X BE → X BF → X CD → X CE → X CF → X

Next are ten categories of discretionary abilities. The difference between some of them can be subtle:

 Swapping the player's choice of: any two cards in row A any two cards in row B any two cards in row C — row is specified, columns are not. Swapping the player's choice of: any two cards in column D any two cards in column E any two cards in column F — column is specified, rows are not. Swapping within the player's choice of rows: left and middle cards middle and right cards right and left cards — columns are specified, row is not. Swapping within the player's choice of columns: top and center cards center and bottom cards bottom and top cards — rows are specified, column is not. Horizontal rotations: rotating right within the row of the player's choice rotating left within the row of the player's choice — direction is specified, row is not. Vertical rotations: rotating down within the column of the player's choice rotating up within the column of the player's choice — direction is specified, column is not. Horizontal rotations: rotating row A in the direction of the player's choice rotating row B in the direction of the player's choice rotating row C in the direction of the player's choice — row is specified, direction is not. Vertical rotations: rotating column D in the direction of the player's choice rotating column E in the direction of the player's choice rotating column F in the direction of the player's choice — column is specified, direction is not. Removals (with replacement): removing a card of the player's choice from row A removing a card of the player's choice from row B removing a card of the player's choice from row C — row is specified, column is not. Removals (with replacement): removing a card of the player's choice from column D removing a card of the player's choice from column E removing a card of the player's choice from column F — column is specified, row is not.

No doubt, someone will extend this into a three-dimensional version.

§9 Number-based 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:

• swapping the highest- and lowest-numbered cards;
• swapping the second-highest- and second-lowest-numbered cards;
• removing the highest-numbered card.
The next abilities, intended for the two-dimensional version, depend on a combination of location and value:
• swapping the highest- and middle-numbered cards of column F;
• swapping the highest-numbered card of row A with the middle-numbered card of row B;
• removing the highest-numbered card in row C;
• removing the lowest-numbered card in column D;
• removing the middle-numbered card in row B.

An ability like "remove the lowest even-numbered card" will fail if all of a player's cards are odd-numbered. 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:

2-6-14-38-W-56-82

but they might reject these:

 2-6-14-38-W-39-82 (W = 38½) 2-6-14-38-39-84-W (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:

• If the first player to claim a sequence has a dirty one, that player is awarded one game point. That player retires, but the other players continue, with the first of the them to claim a clean sequence to be awarded two game points.
• If the first player to claim a sequence has a clean one, that player is awarded three game points, and play ends.

A player who completes a dirty sequence has a strategic choice to make:

• to claim it, and win one sure game point;
• to refrain (at least for the time being) from claiming it, continuing to play and hoping to turn it into a clean sequence, worth more points, before anyone else does.

§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 census-seven game). Still, any card in the sequence is permitted to be frozen.