Phiendish Post-Phestivities Phun (Numerical Puzzle)

[Nigel Graham 2Participant @nigelgraham2]

The diagram below is the principle underlying a board game I played some years ago now. I was reminded by seeing two men playing Ludo in my local cafe this morning.

Play is by conventional counting squares along a track, by die throw, and acting by an instruction you may land on. (As in Snakes & Ladders).

The fiendish  feature is that the board is not a single entity but three physically separate ones you slide together as you decide, to organise the track into partial networks for that session, in one of many N ways..

 

The principle:

Each board is individual, lettered A-C in this model.

Number each edge point 1-8 as shown by Board A.

(Yes I see I should have numbered them more rationally, but never mind, the principle still works. It does not affect solving The Question.)

 

So for example:

The diagram shows A- 5,6,7 will meet B – 1,2,3; A8 > C3; B8 > C1.

C1 forms a dead-end at the A – B joint. All other spurs dead-end at open board edges.

 

Rotating Board A 180º gives A1-B3, A2-B2, A3-B1, A4 – C3, B8-C1.

Etcetera………

 

Each track-section is divided into equal-ish squares, not numbered.

I forget the original details but, e.g.,  each internal cross-road and some inter-node squares carry a direction-instruction (e.g. Left Turn on odd number throws). More tactically, some cross-roads may be left blank for you to decide your route onwards.

Simply chaining the boards end-to-end gives only a modest number of “centipedes” with no loops, so rather uninteresting. Chaining side-by-side, fully or overlapped, gives more options by creating networks with spurs. The only rule is at least one route across or around all three boards.

So lots and lots of potential layouts.

.

Start play by agreeing on any negotiable edge spur as the entrance and if wished, another as the exit, for that instance of playing the game. Fairness dictates navigating all three boards.

E.g., from the diagram, if the square formed by the B1-5 cross-roads with B4-8 instructs “Turn Left ” you reach the edge in this layout, needing more die throws to count you back out.

Or a tween-node square might be blocked (says its printed instruction), so back-track or miss a turn while it is re-opened.

Alternatively, another session’s layout might link that path within a network potentially giving a short-cut past your opponent!

 

 

The Question…..

How Many Layouts Are Possible…?

Is it even calculable, at least without writing an elaborate computer programme?

I posed this at work, using a physical model made of thin card. My audience of about eight were all professional scientists, including a Doctor of Very Hard Sums. They seemed unsure.

 

Where did I find this brain-teaser?

It is a map game invented by two American cavers, refined with help from friends, and if I remember correctly, they called it Krazy Kaverns. (Yes I did notice omitting the “n” might be appropriate!)

They based it on cave surveys (“maps”) with appropriate obstacles or advancements sprinkled around it, and it is presumably, rightly copy-righted; but the basis could for example be roads and villages – land on “The Crown”, stop one turn for lunch – or countryside footpaths.

 

Happy Sum-smithing!

 

 

 

 

 

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