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Subject: A class of orderings for 504 games with some nice properties rss

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Sophia Lechner
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In what order shall we play our 504 games? Let's back up a step. What would be a set of nice properties for such an ordering to have? It's subjective, to be sure, but there's a sense in which I'd like the ordering to maximize variance from game to game, and I'd also like it to be easy to figure out the next game to avoid complex lookups.

I suggest the following properties (the first two for variance, the third for ease of use):

1) Each set of 9 games (the first 9, the next 9, etc)
1a) Should contain each module once in position I
1b) Should contain each module once in position II
1c) Should contain each module once in position III
2) Each set of 72 games should contain
2a) Each ordered pair of modules once in positions I and II.
2b) Each ordered pair of modules once in positions I and III.
2c) Each ordered pair of modules once in positions II and III.
3) In each set of 9 games, all but the first should be generated from the previous game by going to the next module in each position (wrapping from 9 to 1 as needed).

Property 2 is surprising difficult to satisfy (any two of the subconditions are trivially easy, but not three jointly). However, the following scheme accomplishes it:

3 4 5 6 7 8 9 2
6 2 9 3 8 7 5 4
7 6 2 4 3 9 8 5
4 3 8 2 9 5 7 6
9 5 4 8 2 6 3 7
5 9 3 7 6 2 4 8
8 7 6 5 4 3 2 9
2 8 7 9 5 4 6 3


For each of the 56 cells in grey (left-to-right, then top-to-bottom), play 9 games chosen as follows:

Play game 1xy, where:
x is the highlighted number on top of the column
y is the number in the grey cell

Play the game formed by adding one to each module number from the previous game (if the module was 9, replace it by 1).

Repeat until you've just played a game with 9.I, then go to the next grey cell.

For example, the first game is 136 (6 in the grey cell, 3 above), the next game is 247, then 358, 469, 571, 682, 793, 814, and 925. The next game is 142, then 253, etc...

This satisfies all of the properties above. I am convinced by extensive search in Excel (though I have no formal proof), that this is the unique ordering with these properties, discounting the following trivial modifications:

You can reorder the grey rows (not the yellow one) freely
You can reorder the columns freely
For each set of 9 games, you can start with any game in that set (though always starting with I.1 has the additional nice property that you always go nine games before replaying a module in position I).

[edit: fixed and expanded example]
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Byron S
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JemL wrote:
For example, the first game is 136 (6 in the grey cell, 3 above), the next game is 249, the next is 351, and so on to 925. The next game is 142, then 253, etc...

Should this be 136, 247, 358, ... ? Otherwise that 9 is really throwing me off.
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Sophia Lechner
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runtsta wrote:
JemL wrote:
For example, the first game is 136 (6 in the grey cell, 3 above), the next game is 249, the next is 351, and so on to 925. The next game is 142, then 253, etc...

Should this be 136, 247, 358, ... ? Otherwise that 9 is really throwing me off.
Argh, yes. Thank you, fixed now.
 
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Brian Dysart
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Wonderful high nerdery. Bravo.
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Joseph Anderson
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Interesting, If I were trying for diferent experiences would have a slightly different set of criteria:
1. in every set of 3 games play with every module.
2. in every set of 9 games play with every module in position I, II and III.
3. in every set of 72 games play with every combination of module I and II
4. in every set of the first 84 games play every combination of modules. (not caring about position)

Working out a scheme for that would also be tricky but I think I wouldn't care about this too much. I'm likely to use some different selection method.

(the importance of the 4th criteria is that you will never repeat a combination of modules in the first 84 games)
 
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Peter Hendee
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knaves wrote:
1. in every set of 3 games play with every module.
2. in every set of 9 games play with every module in position I, II and III.


I used the above criteria for selecting nine games that would give me a flavor for the system.

Edit: Now that I have seen the rules I would also want to make sure I triggered lots of rules that vary by game such as playing all 5 map configurations and seeing as many components as possible.
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Russ Williams
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I've been pondering that sort of approach as well, inspired by all-pair testing from software testing methodology.
https://en.wikipedia.org/wiki/All-pairs_testing


But we are also tempted by the idea of simply selecting each game uniformly randomly (from among the combinations we've not yet played).
 
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Friedemann Friese
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If you want to explore the 504 worlds best, it is of course helpful to first play 9 different games to have each module on one of its 3 positions. And yes different combinations of single modules is helpful, but theres is some more:
Spoiler (click to reveal)

- Transport(1) together with production (8) has to be played, because you produce the good, you have to transport afterwards
- Racing (2) on TOPI without residents has to be one, because this has a special ending condition of reaching 9 cities and getting extra VPs for additional spaces.
- Be sure to use all different board setups.
- Exploring(5) from only one center city is a new experience (both very different with or without residents)
- Military(4) on TOPI or TOPII with racing(2) is something new, because fighting is includes in the moving your residents phase, means you can rush.
- transport TOPIII with street building and you transport truck becomes a street builder.
- Military(4) with Production(8) creates cities where the players are not allowed to fight in.
- You need a lot of games with Privileges(3) to see all the different cards in use
- racing on TOPIII has a chart of 7 different ways (depending on TOPI) to score one time VPs in the game, but especially with Share(9) TOPI this will be transferred to Income.
- Privileges TOPII with Majorities TOPI creates a turn order depending on VPs, but you have to pay the taxes depending on the highest income.
- Military with almost everything else changes a lot, because there is normally only one color allowed per hex.
- Exploring with Privileges factory building and the special rule to move as many residents as you like active in a new explored space is worth a try
- Street building (6) with transport(1) is tricky because you build the streets to have a faster movement for your transport truck.
- Majorities TOPI can be so much different depending if the others modules generate fast or slow money. Sometimes almost all residents and settlement have to be removed in each scoring round.
- Shares (9) TOPII or TOPI together with majorities TOPIII is very interesting because of the last impact of the final majority scoring.


Just to say that you can not solve this only with mathematics or statistics.....

Have fun

fri504emann
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Russ Williams
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friedemann wrote:
Just to say that you can not solve this only with mathematics or statistics.....

But formal methods are not truly helpless here; good software testing methodology will cover all these interesting cases...

BTW this is the "hot Essen game" which has most caught my interest in years! I'm very much looking forward to 504.
 
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Dan
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Wow, friese is like a mad scientist! 504 is definitely on the wishlist.
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Kim Choy
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You know, Sudoku are a lot more interesting if you don't fill in all the numbers for us beforehand.
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Sophia Lechner
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knaves wrote:
Interesting, If I were trying for diferent experiences would have a slightly different set of criteria:
1. in every set of 3 games play with every module.
2. in every set of 9 games play with every module in position I, II and III.
3. in every set of 72 games play with every combination of module I and II
4. in every set of the first 84 games play every combination of modules. (not caring about position)

I like your criterion 1 a lot! Better than my criteria 2. Now I'm wondering if these are possible. I suspect they are...not sure I have time soon, but methods for making this happen are going to be wandering around my head. (Even the original stronger version of your #4 may be possible...hmm). Thank you for the inspiration. Will follow up later.
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Sophia Lechner
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friedemann wrote:
If you want to explore the 504 worlds best, it is of course helpful to first play 9 different games to have each module on one of its 3 positions. And yes different combinations of single modules is helpful, but theres is some more:
Spoiler (click to reveal)

- Transport(1) together with production (8) has to be played, because you produce the good, you have to transport afterwards
- Racing (2) on TOPI without residents has to be one, because this has a special ending condition of reaching 9 cities and getting extra VPs for additional spaces.
- Be sure to use all different board setups.
- Exploring(5) from only one center city is a new experience (both very different with or without residents)
- Military(4) on TOPI or TOPII with racing(2) is something new, because fighting is includes in the moving your residents phase, means you can rush.
- transport TOPIII with street building and you transport truck becomes a treat builder.
- Military(4) with Production(8) creates cities where the player are not allowed to fight in.
- You need a lot of games with Privileges(3) to see all the different cards in use
- racing on TOPIII has a chart of 7 different ways (depending on TOPI) to score one time VPs in the game, but especially with Share(9) TOPI this will be transferred to Income.
- Privileges TOPII with Majorities TOPI creates a turn order depending on VPs, but you have to pay the taxes depending on the highest income. Special
- Military with almost everything else changes a lot, because there is normally only one color allowed per hex.
- Exploring with Privileges factory building and the special rule to move as many residents as you like active in a new explored space is worth a try
- Street building (6) with transport(1) is tricky because you build the streets to have a faster movement for your transport truck.
- Majorities TOPI can be so much different depending if the others modules generate fast or slow money. Sometimes almost all residents and settlement have to be removed in each scoring round.
- Shares (9) TOPII or TOPI together with majorities TOPIII is very interesting because of the last impact of the final majority scoring.


Just to say that you can not solve this only with mathematics or statistics.....

Have fun

fri504emann
Just when I thought my anticipation for this game couldn't be increased, here you are! Looking forward to these so much! Fortunately, if I stick to the plan outlined here, I'm certain to experience all of your suggestions within my first 72 games.
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Jack Spirio
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This just makes me so excited for this game, even dough I was already total excited about it
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Joseph Anderson
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I really like the idea of trying to experience every aspect of the system but I think that I'm going to gravitate towards specific aspects.

I'm especially intersted in Modules 1, 3, 5 and 9. The other modules interest me as well but mostly in relation to those 4. So do I play around with the modules that most interest me or do I try for the most diverse play experience...
The answer is probably somewhere in the middle.
 
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Sophia Lechner
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knaves wrote:
Interesting, If I were trying for diferent experiences would have a slightly different set of criteria:
1. in every set of 3 games play with every module.
2. in every set of 9 games play with every module in position I, II and III.
3. in every set of 72 games play with every combination of module I and II
4. in every set of the first 84 games play every combination of modules. (not caring about position)

Working out a scheme for that would also be tricky but I think I wouldn't care about this too much. I'm likely to use some different selection method.

(the importance of the 4th criteria is that you will never repeat a combination of modules in the first 84 games)


Progress report... here are 12 games to get you started that satisfy your properties 1 and 2, and I've determined that they can be (non-uniquely) extended to an order of games that satisfies your properties 1, 3 and 4 (the extended "every set") version. I'm still unsure as to whether I can extend them in such a way as to also satisfy property 2. However, this set does also satisfy

5. The first 12 games include each pair of modules (ignoring position) exactly once.

and a less-important-but-why-not

6. Follows the rulebook recommendation that the first three games be 123, 456, and 789.

123 456 789
618 375 942
267 834 591
147 258 369

I don't think there will be a way to cleanly generate such a list as the result of a short algorithm, though. Getting a list of 84 games all different ignoring position was already a lot of trial-and-error, and finding such a list that (started, anyway) to satisfy properties 2 and 5 was even more.
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Joseph Anderson
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On one hand I really love what you're doing, I love this approach. After your initial post, I did in fact spend a while, coming up with my set of criteria and looking for ways to resolve it.

However, having now played a few games by forum, I am really starting to see that the strength of the system is not just the variety in big changes but the variety in small changes to the module set up.

If I play 536 and love it but we felt that we really miss played it then we might repeat it. or try 653 or 538 shortly afterwards.

The ability to play games that are similar but with significant differences to recent games is as significant to me as the variety possible in the first 72-84 games.

Please don't take this as me not appreciating what you are doing here. You're doing fantastic stuff. (and I am a bit of a math and stat geek.)
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Sophia Lechner
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Aw, thank you. Yeah, you're certainly right...there's so much variety in changes that are minor by these criteria that it's not really meaningful.

I mean, we just played 789...imagine 978; suddenly we don't have player colors, are trying to amass the most valuable (in $) collection of stocks by the end of the game, and stock values are changing based on how many of each terrain type the companies have relative to the other companies, but the companies are restricted in their resident purchases by the number of plants they've built. It's not remotely the same game.

So I get that...I didn't do this because I thought it would make playing 504 more fun, I did it because it was fun to do!
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