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Subject: New combinations under our current assumptions rss

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Darren Nakamura
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So, we haven't seen the rules for Nomads yet, but assuming that it is adding four new main location types (not counting the nomad tents--however those will end up working) and three new Kingdom Builder cards, that brings us to twelve total boards and thirteen total cards.

So the new total number of different starting setups would be:

P(12, 4) x 2^4 x C(13, 3) = 54,362,880

Where it used to be:

P(8, 4) x 2^4 x C(10, 3) = 3,225,600

And as mentioned before, that's not even taking into account the nomad tents, since we don't know yet how those will be incorporated. If they are just shuffled and randomly assigned to the new boards, then that will increase the possible starting setups even further (though in a manner I'm not exactly keen to attempt to calculate).
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Russ Williams
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Calculating the increase in possible setups when using Kingdom Builder: Capitol is left as an exercise for the reader!

(For each Harbor or Oracle in play, a capitol can go on either of that board's 2 castles.)
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Tyler Somer
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Dexter345 wrote:
So the new total number of different starting setups would be:
P(12, 4) x 2^4 x C(13, 3) = 54,362,880

Where it used to be:
P(8, 4) x 2^4 x C(10, 3) = 3,225,600


The board orientations are (2^4), but we repeat each and every possibility by rotational symmetry, so divide this by 2 to get (2^3).
That is to say,
AB
CD
is the same as
DC
BA
under a rotation of 180 degrees.
(One could just as easily argue that this affects the permutation calculation, and not the orientation calculation. In either case, the division by 2 must occur at some point, and the final result is the same.)

The numbers are:
P(12,4) = 11,880
2^3 = 8
11,880 x 8 = 95,040 Board Set-ups.
C(13,3) = 286 Card Selections
95,040 x 286 = 27,181,440 Complete Game Set-ups.

The original numbers were:
(1,680 x 8) x 120 = 13,440 (Boards) x 120 (Cards) = 1,612,800 (Games)

One additional note on the calculations:
For anyone who is interested, once you have a selection of 4 quadrants, these can be arranged in 192 Board Set-ups.
(P(4,4) x 2^3 = 24 x 8 = 192)
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Tyler Somer
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russ wrote:
Calculating the increase in possible setups when using Kingdom Builder: Capitol is left as an exercise for the reader!

(For each Harbor or Oracle in play, a capitol can go on either of that board's 2 castles.)


I'll get to work on that soon...
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Tyler Somer
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russ wrote:
Calculating the increase in possible setups when using Kingdom Builder: Capitol is left as an exercise for the reader!

(For each Harbor or Oracle in play, a capitol can go on either of that board's 2 castles.)


Calculating the selection of quadrants with the base game:
Case 1 - Games with neither the Harbor nor Oracle: C(6,4) = 15
Case 2 - Games with the Harbor and not the Oracle: C(6,3) = 20
Case 3 - Games with the Oracle and not the Harbor: C(6,3) = 20
Case 4 - Games with both the Harbor and the Oracle:C(6,2) = 15
Verification: all games: C(8,4) = 70 (=15+20+20+15)

Calculating the board set-ups with Capitol included:
Case 1: 15 x 1 x 192 = 2,880
Case 2: 20 x 2 x 192 = 7,680
Case 3: 20 x 2 x 192 = 7,680
Case 4: 15 x 4 x 192 = 11,520
Total: 29,760 board set-ups
Factor in the card choices: 3,571,200 game set-ups.

Selecting the quadrants with the upcoming expansion:
Case 1: C(10,4) = 210
Case 2: C(10,3) = 120
Case 3: C(10,3) = 120
Case 4: C(10,2) = 45
Verification: C(12,4) = 495 (=210+120+120+45)

Calculating the board set-ups with Capitol included:
Case 1: 210 x 1 x 192 = 40,320
Case 2: 120 x 2 x 192 = 46,080
Case 3: 120 x 2 x 192 = 46,080
Case 4: 45 x 4 x 192 = 34,560
Total: 167,040 board set-ups
Factor in the card choices: 47,773,440 game set-ups.

Thanks for suggesting this diversion, Russ! For brevity, I have not illustrated all the calculations...
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Darren Nakamura
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JackBurr wrote:
The board orientations are (2^4), but we repeat each and every possibility by rotational symmetry, so divide this by 2 to get (2^3).
That is to say,
AB
CD
is the same as
DC
BA
under a rotation of 180 degrees.
(One could just as easily argue that this affects the permutation calculation, and not the orientation calculation. In either case, the division by 2 must occur at some point, and the final result is the same.)


Good call; I hadn't considered that.
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KC Skedzielewski
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But wait,

AB
CD

Is not the same as

DC
BA

in terms of game play as if you always face the locations upwards, those would be vastly different layouts. So I dont know if you can just reduce the set of variants like that.
 
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Tyler Somer
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kcskedz wrote:
But wait,

AB
CD

Is not the same as

DC
BA

in terms of game play as if you always face the locations upwards, those would be vastly different layouts. So I dont know if you can just reduce the set of variants like that.


If you indeed limit yourself to having the locations always facing upwards, then you only have 24 possible arrangements once you have selected a set of four quadrants. (4! = 24) However, you have failed to consider that each quadrant can be oriented in one of two ways, so this leads to 2^4 = 2x2x2x2 = 16; then 24x16 = 384.

Let me use some different letters to illustrate that 24x16 = 384 is incorrect, while 24x16/2 = 192 is correct:

Rewrite
AB
CD

as

bd
mn

and notice that the player across the table would see this as
uw
pq
...

Now consider
DC
BA

which becomes

nm
db

but now turn each quadrant around by 180 degrees:

uw
pq

and now the first player sees the board as the second player did in the first arrangement.

As I mentioned in an earlier post, above, one could argue that the permutation calculation must be adjusted by way of a division by 2, or the orientation calculation must be adjusted. But either way, the division by 2 must occur at some point!

You have chosen to isolate the permutation calculation (24). Don't stop there! Time to look at the orientations.
 
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Russ Williams
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JackBurr wrote:
It appears that of the three cards in this expansion, two are replacements/updates,

Hmm?

The rules say that there are 3 new cards (Families, Shepherds, Ambassadors), plus 2 replacements of existing cards (Workers, Merchants).
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Tyler Somer
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russ wrote:
JackBurr wrote:
It appears that of the three cards in this expansion, two are replacements/updates,

Hmm?

The rules say that there are 3 new cards (Families, Shepherds, Ambassadors), plus 2 replacements of existing cards (Workers, Merchants).

Thanks for the clarification. I've therefore deleted my incorrect speculations.
 
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