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A Board Game Problem - How Many Possible Playing Routes?

A typical board track-game has a single, rigid route of numbered squares, controlled by die throws, with certain actions given in some squares - e.g. "Snakes & Ladders".

One track from start to end even if you must return somewhere, miss a turn or advance N squares, etc. as ordained when landing on those given points


Now set the game on THREE separate, identically-sized, rectangular boards of 3:2 proportions; EACH marked off into 6 squares by a longitudinal centre-line and two cross-lines at thirds length.

Label the boards (for this question on the game's geometry) A, B and C. Label each meeting-point of line and edge, 1 - 6, consistently on all 3 boards so every "port" is unique, A1 to C6.

Do that with the individual board in landscape mode; 1 on the left-hand end, number clockwise so the RH end is 4, round to 6 at bottom left.  All 3 boards the same apart from identity-letter.

 The track is drawn on these lines such that when one board is set against the others for play, at least one route is feasible from board to board. The track squares are NOT numbered: that would not work, as a little thought shows.

You could just put the boards end-to-end (A4 - B1, B4 - C1) or (A1 - C4, C1 - B1), say, but the game would be rather dull, as the track is then just a spine. It is better to join the boards to give a layout with at least one loop as well as dead-ends (on the board edges) - which you do use as well as the continued line, though I can't recall how - whether, e.g. a direction-choice set on its junction, controlled by the number thrown, or simply by odd/even or particular-digit.

A quick sketch shows the three boards side-by-side gives two 1 x 1 "square" loops with a common centre-path, also readable as a 2 x 1 rectangle with central cross-path, surrounded by 10 dead-ends. And of course, you can turn any board 180º and it will still give that layout but change the action-squares' relative locations!

Nor do I recall the starting-point -whether the designer designated a certain square that would appear where your layout for the session puts it relative to the other two boards, or if you are free to set a sensible point, perhaps one of the dead-ends on the edge of the whole layout. Anywhere would lead to a cross-roads within a few squares.

SO.....

How many possible, playable layouts, including simple end-to-end spines, remembering every one of the 18 board exit-points is unique?

I made a basic, line-only model of the game and gave it to several professional physicists and mathematicians... They suggested a few numbers like 36, then thought, "Oh, err, it's harder than that!" Any ideas anyone?

BTW this is about a genuine game I have played!   

Posted - January 8, 2018
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Responses


  • I would have to make a set of boards, and then make diagrams of each arrangement, and count the possible pathways based on the possible sequences of throws of the dice.
    6 possibilities for a regular dice.
    17 lines for each board, but depending on how they connect, if the edge lines that touch are counted as only one, the number of possible lines across the three boards varies for each game.
    Are there rules or levels which determine the arrangements?
    I wonder if the quick sketch looks like an H pattern?

    Does the game have a name?
    From which country does it originate?
    I'm guessing perhaps the Middle-East.
    I would love to look it up online.



    This post was edited by Benedict Arnold at January 8, 2018 7:43 PM MST
      January 8, 2018 7:41 PM MST
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  • 3719
    Ah, you don't need to go into the added complexity of die throws - just the board geometry!

    I may have been better not mentioning the die, as it does not affect the geometry puzzle. I added it to show the cul-de-sacs ending at outer board edges were not left abandoned, but are used to increase the game's variety still more. One you've laid out the three boards as you wish the basic play is as on any similar track game, but with the spice of many versions of the geometry, including internal circuits.

    As I remember, the game is or was called Krazy Kaverns.

    It was invented ooh, 20 years ago now? by two American cavers, and refined with ideas from their friends. I've only seen it once, owned by friends who I think had bought it in America, though I'm not sure about that.

    Its artwork is based on standard cave-survey conventions, and the whole game is gently self-deprecating humour by and for cavers; an in-joke really. It's ideal for playing in the caving-club cottage when the pouring rain makes your intended cave too risky and hill-walking horrible! Which is basically where and how I encountered it.  

    I deliberately omitted its name and nature above so it didn't distract from my real question, on how many ways to link three uniquely-numbered but equi-sized "hexapedes" so at least one "foot" meets one on its neighbour.  When I asked my work colleagues I used a model comprised of three pieces of plain card cut to the right proportions, with just the three ruled lines, letters and numbers on each.

    An offset H-pattern is certainly possible, with several ways to arrange the boards to meet the criteria in the same overall H shape, but the paths won't form loops (or circuits). The quick sketch I made gives a straightforward rectangle with long-sides in full-length contact, so 2 ports on one board meeting 2 on the next, and this does give loops.

    E.g. A,C,B so

        A6 & A5 meet C2 and C3; and C5 and C6 meet B2 and B3.

        Flip C round and A5 & A4 meet C5 & C6, then C2, C3 meet B3, B2... respectively.

    So that is two very different iterations of the (2-adjacent squares with circumscribing rectangle) above; because the various instructions now lie on different routes around the loops of track. 

    The only rule is that at least one exit on each board enters a path on the next, but the proportions are such that simple neatness in placing them on the table should do that. There are no "levels" as such but the game is much more interesting when you link the paths to form loops. Although simple end-on chains, and L-  T- and H-patterns, are allowable and part of the geometry puzzle, they are less fun to play.

    ('Loop' is self-explanatory here, even though I've condensed the geometry to squares and rectangles for the question; but is a term actually in cave-surveying, even if the passages wiggle all over the place before re-connecting somewhere else. Measurement-quality is assessed by "loop misclosure".)

    I simplified the model to purely diagrammatic to concentrate on the combinations problem. The fictional cave's passages depicted in the game's artwork wiggle as real ones do, for aesthetics and similarity to real surveys, but their meetings with the boards' edges all obey the basic 3:2 geometry.

    I've been puzzling how to represent one sub-board on here but text-editors are not good for that!

    I've just tried looking but "Goggles" corrected the spelling to Cs and offered a computer fantasy game nothing to do with Krazy Kaverns with Ks. 

      January 8, 2018 9:26 PM MST
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