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!