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I was wondering someone could help. I've developed a board game which is made up of six, large hexagonal board pieces, which can be arranged in any order, and with any rotation/arrangement of sides within that overall layout.

I was playing with a friend who made a laughable attempt to calculate how many possible layouts there could be, and got me thinking I literally have no idea where to start on the maths of it. Hence I'm here.

So for example the six could be laid out in a single row, or could be bunched together like in the picture... and then within that, each tile can be rotated.

The number must be astronomical, but how would I (or you!) figure it out?

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Many thanks in advance and sorry to trouble you all.

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  • $\begingroup$ Astronomical numbers are quite small, by mathematical standards. $\endgroup$
    – Gerry Myerson
    Commented Jul 21, 2020 at 23:03
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    $\begingroup$ oeis.org/A000228 may give you a start. $\endgroup$
    – Gerry Myerson
    Commented Jul 21, 2020 at 23:06
  • $\begingroup$ Just a thought: if you think of the dual, I think you are looking at number of ways of connecting the edges of 6 vertices of degree 6, so that the final graph is connected. Since you have numbered the sides, you would also need to number the edges. Maybe looking at the configuration model (en.wikipedia.org/wiki/Configuration_model) is useful. $\endgroup$
    – gmath
    Commented Jul 22, 2020 at 21:46

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