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Help with statistics

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Anonymous

I am pondering a random tile set-up similar to Catan, but with squares instead of hexes. If anyone has any thoughts on how to determine some things mathematically, that would be swell!

There are 36 tiles, which will be arranged in a 6x6 square. Each tile has up to four (and as few as two) Connectors, which are centered on the sides of each tile. (A tile could have connectors on the North, South, East and/or West side.) When the Connectors of two tiles are placed next to each other (a North placed next to a South, or an East placed next to a West) a Link is formed. Connectors on the edge of the 6x6 square can not form a Link (since it is impossible for them to be next to another tile).

Here is the breakdown of how many tiles have how many Connectors.
12 tiles have 4 Connectors each.
20 tiles have 3 Connectors each.
4 tiles have 2 Connectors each.

I am trying to determine 3 things, based on these tiles being placed randomly in a 6x6 square;
1- What is the maximum number of links that can occur?
2- What is the minimum number of links that can occur?
3- What is the average number of links that will occur?

By drawing a grid out, I came-up with the following:
-There are 60 tile-side to tile-side edges where a Link could possibly form.
-The 20 tiles with 3 Connectors means there are up to 20 Links that can not be formed.
-The 4 tiles with 2 Connectors means there are up to 8 Links that can not be formed.
-Subtracting 20+8 from 60, leaves you with a Minimum of 32 Links that will be formed by any random placement.

Am I anywhere close?

Rick-Holzgrafe
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Re: 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? :)

Anonymous
Re: Help with statistics

Rick-Holzgrafe wrote:
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.....<> Make sense, or am I babbling? :)

Great stuff Rick!
(Forgive me for jumping into the conversation)

I can use the same math for my dungeon tile laying game. I've been using a 5 by 5 grid, with the starting tile being in position 3 from the left. (The southmost edge of the board). I had an issue with too many deadends, and your analysis of what CaptainBob's situation helps me as well.

Thanks a million!

Falloutfan.

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