Help with statistics

There are 24 tile-sides which cannot form links (regardless of what is on the tile) because they are on the edge of the 6x6 square.

There are 28 tile-sides which cannot form links (regardless of where they are placed) because they do not have connectors.

There are 144 tile-sides in all; if all of them participate in Links (they can't, of course) then there would be 72 Links.

If every tile had four Connectors, there would be 24 unconnected ones (the ones on the edge of the 6x6 square), which means that 12 possible Links are unrealized due to edge positioning. This reduces the maximum number of Links to 60 (which agrees with your number, Cap'n).

There are 16 tile locations that have EXACTLY one outside edge. Assume we place a 3-Connector tile in each of these locations, unconnected side outwards. That leaves four more 3-Connector tiles unplaced.

Assume that all of your 2-Connector tiles have right angles (their Connectors are on adjacent sides). Place these at the corners, connectionless sides outward.

We now have 4 remaining internal edges that have no connectors. Assume we pair these, so that all remaining Connectors participate in Links. That means that 2 potential Links remain unlinked. 60 minus 2 equals 58 Links. I think this is the theoretical maximum.

If all of your 2-Connector tiles are "straight-through", and we place them at the corners as before, then four more potential Links are not formed. Your maximum becomes 54 Links. My guess is that you have two of each, so the maximum is actually going to be 56 Links.

I think your calculation for the minimum is correct, and I'll add that with random placement, it's unlikely that you'll get the minimum.

What's the average? I think that's harder to compute. But look at it this way: there are 28 connectorless edges, which is about 1/6 of the total of 144 tile edges. With a random distribution of edges, maybe 5 connectorless edges will wind up facing outward; the rest, about 23, will be internal. That's 23 connectorless edges out of 120 total internal edges, or about 1 in 6 internal edges without connectors. This ratio suggests it's unlikely that very many of those edges will face each other, so most of the 23 also represent lost Link opportunities. So I think on average you'll miss maybe 15 to 20 possible Links, leaving you with an average of 40 to 45 Links.

Make sense, or am I babbling? :)