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Winter Kingdom Builder Math

M. C. DeMarco
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In the Winter Kingdom comments, someone remarked on the large size of the modular board in Winter Kingdom. The Kingdom Builder board is 10x10x4 (400 hexes), while the Winter Kingdom board is a 5x5x5 hex (61) x 7 = 427 hexes, which is an increase of only 7%. When you consider that much of this extra space is occupied by the usually inert new "village" locations (14 out of the 27 extra hexes), it's not really much of a change. However, the sheer number of possibilities has gone up quite a bit because of the number of modules to the board (up to 7 from 4) and the increased number of possible positions of the board sections, including flipped to their new back sides.

A Kingdom Builder modular board section has two possible positions, which we can call right side up and upside down based on the orientation of the location hexes. The base game comes with 8 board sections. You choose four of these sections for the base game, in order, so there are 8 x 7 x 6 x 5 (1680) possible ordered sets of boards. Each modular board has two positions, so within each set there are 2 x 2 x 2 x 2 (16) possible layouts, for a total of 26,880 possible game boards. However, for the purposes of the game, a game board is the same as the upside-down version of itself, so we need to divide this total by two to handle the symmetry, leaving 13,440 boards. Each of the four expansions comes with four more sections, so if you have all the expansions (as in the bigger of the Big Boxes), the number of possible boards is 24 x 23 x 22 x 21 x 16 / 2, or just over 2 million possible board layouts.

A Winter Kingdom modular board always uses all 7 board sections, so the only question is the order: there are 7 x 6 x 5 x 4 x 3 x 2 x 1 (5040) possible orders. For each modular board you must also choose 1 of 2 sides (2 x 2 x 2 x 2 x 2 x 2 x 2 = 128) and one of six orientations (6 x 6 x 6 x 6 x 6 x 6 x 6 = 279,936) per side. There is, however, a sixfold symmetry to the board, since rotating it around the center does not make a difference to the game, so we divide by 6 (instead of 2) as in the Kingdom Builder case. The total is 5040 x 128 x 279,936 / 6, or 30,098,718,720---a bit over 30 billion possible boards.

There are a few threads at BoardGameGeek that calculate both the number of Kingdom Builder boards and the possible base game configurations including Kingdom (goal) cards: a thread on an image, a bad guess in 2011, and a later 2015 thread. For the base game, there are 10 x 9 x 8 / 6 = 120 options for combinations of the goal cards. Multiplying by the boards, there are 13,440 x 120 = 1,610,000 possible Kingdom Builder games.

Once again Winter Kingdom kicks it up a notch, with 18 goal cards (18 x 17 x 16 / 6 = 816 combinations. There's also one economy card out of 8, and zero or one twist cards out of 8 (zero is recommended for beginners), for a total of 816 x 8 x 9 = 58,752 possible card setups. Multiplying by the boards, we get 30,098,718,720 x 58,752 = about 1.8 quadrillion possible games.

In Kingdom Builder, ability tiles are associated with the boards, while in Winter Kingdom abilities are an extra deck of 25 cards from which each player is dealt 5. This amounts to 25 choose 5 times 20 choose 5 (times 15 choose 5 (times 10 choose 5)) possible hands of 5 cards for 2, 3, or 4 players. That's 53130 x 15504 (x 3003 (x 252)), or between 824 million and 623 trillion combinations. If you consider only the hands drawn and not the order (who gets them), there are between 412 million and 26 trillion combinations, for a total of between just under a septillion combinations for 2 players, up to 46 octillion combinations for four players.

About the Boards

Board Game: Winter Kingdom
Board Game: Winter Kingdom

It's not a graph, but I gave the Winter Kingdom boards a good look-see in the process of making a Winter Kingdom board generator along the lines of my old Kingdom Builder board generator:

External image

I came up with a scheme for naming the boards, for discussion below or to help generate or refer to particular layouts. (With no distinct locations, this is more difficult than in Kingdom Builder.) Orient the board with the castles and villages upright in order to find the top row, and then count the number of hexes of the first terrain type in that row. Add a second terrain type if the first is ambiguous. Use upper and lower case to distinguish the front and back sides. The boards in this scheme are:

3 Fell / 2 ice 3 grass
3 Forest / 2 grass
4 Fell / 1 mountain 3 forest
3 Flower / 1 grass
3 Tundra / 1 forest
2 Ice 3 Snow / 1 snow
1 Mountain 4 Tundra / 1 flower

You can abbreviate them thus:

3Fe / 2i3g
3Fo / 2g
4Fe / 1m3fo
3Fl / 1g
3T / 1fo
2I3S / 1s
1M4T / 1fl

The orientation of the board can be referred to using degrees of rotation to the right: 0° is upright, 60° is angled slightly to the right, and the next four positions are 120°, 180° (upside-down), 240° and 300°. If it's less confusing you could go negative, with the same positions being 0°, 60°, 120°, 180°, -120°, and -60°.

You can find a different, incomplete naming scheme in this thread.

The smallest size for regular terrain (not ice, mountains, or locations) is 3 hexes; there are a bunch of 3 hex regions on 2i3g. The largest size for regular terrain is 9 hexes (tundra on 2g); there are also several 8 hex regions (e.g., grass on 3T, tundra on 3Fl). Regular terrain of size 3 or 4 hexes is usually balanced out by another such region on the same board (but see 2i3g for exceptions); as a result there are few regions of 5 hexes (e.g., snow on 3T, snow and fell on 2i3g).

Ice and mountains can appear in single hex size (e.g., on 1s and 1fl), and mountains also occur in 2 hex chunks (also on 1fl). Ice and mountains may fail to occur (on 1m3fo and 2I3S, respectively), or may occur in regions of more than 10 hexes (e.g. the circular river on 2i3g and the transcontinental range on 3Fe), but the regular terrain types appear on every board, either in one large chunk or two smaller chunks.
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Mon Jul 19, 2021 11:06 pm
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Getting Better and Better

M. C. DeMarco
United States
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Today's reproduced graph is from a relatively recent pair of Reddit posts by u/backlogplayer, meant to demonstrate that games keep getting better over time using some notion of the ratings for the best game of the year.

The initial post (image link) used average rating rather than geek rating, with no correction, and ran into the usual issues of obscure, highly-rated games percolating to the top for some (though not all) years.

The updated post (image link) switched to geek rating, and commenters then noted that the graph is more about ratings inflation over time than an actual change in game quality, and also that BoardGameGeek only started in 2000, so the earlier years are qualitatively different. There's also an outlier in 2018 about which the OP speculates; it's actually there because the source data didn't include 2018 at all, though BGG does include expected releases and their expected date of publication, and though people with playtesting access rate those games (giving them a rating to be picked up), the geek rating will be low because of low rating counts.

Anyhoo, here are some basic reproductions of the two graphs, in case you don't feel like clicking through above:

From gallery of fiddly_bits

From gallery of fiddly_bits

I spiffed one up with the actual game names:

From gallery of fiddly_bits

Some commenters made the legitimate point that the best game of the year is itself an outlier, so I expanded the graph to the best 25 games of each year:

From gallery of fiddly_bits

I started at 100 games and worked my way down; the pattern didn't change significantly. Note that 2018 remains an outlier here, probably because it takes more time for the 2018 games to percolate into people's hands and ratings than my February 2019 dataset allows for.
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Thu Apr 11, 2019 3:37 am
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Money Approaching Mouth

M. C. DeMarco
United States
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I've been puttering away reproducing some old graphs of BGG data with my more up-to-date dataset. Today's graph was inspired by a Reddit post by Alan G. Wilson (u/zelph104), made pretty by an online data display tool called Tableau.

I have not recreated the prettiness, though it could be done by importing the data into Tableau. Instead I recreated the graph of how often BGG users have put their money where their ratings are: the ownership levels of highly rated games. Unlike the original chart, mine has a rating floor in the 5's somewhere due to my data, and I've also reproduced an ownership floor in the source graph that doesn't exist in my data.

Color in the original graph corresponded to rating (the y-axis) and so wasn't interesting. I perked it up with fan counts (first graph) and year of publication (second graph, which is also restricted to the modern board gaming age for the purpose).

From gallery of fiddly_bits

From gallery of fiddly_bits
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Mon Apr 8, 2019 7:52 pm
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Day of the Data

M. C. DeMarco
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This post is a bit of a cheat, because the graphs are all in my account at Kaggle. They're also a bit wee and uninformative, but they do include fans (if you expand the display to 25 columns, anyway), because Kaggle automatically graphs the columns of your datasets. Here's a screenshot:

From gallery of fiddly_bits

My dataset is a new, February 2019 version of this dataset of the top 5000 games at BGG, fetched using somewhat revised code.
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Thu Feb 7, 2019 10:27 pm
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Return of the Recs

M. C. DeMarco
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Yesterday BGG announced a new recommendation system, which I heard about on reddit despite having visited BGG since the announcement. (Some might say that has something to do with the desperate need for the site overhaul to reach the front page of BGG before new upstarts take over their market, but they don't generally say that with graphs so it's off-topic here.)

Though they say they've "reinstated" recommendations, I believe (based on hints dropped by those who remember it) that the previous recommendation system was user-specific and used your own ratings of multiple games to suggest new ones---like the user correlation tool I discussed last time or the personalized recommendation tool at Ludoj. In fact, many of the comments in the thread are about Ludoj instead, and whether people liked its recommendations. (I thought they were fine for me, except that I'd played most of the games and couldn't get the feature to exclude my played games working.)

The new system is just a general graph calculation of popularity for the particular game you're looking at, based on ratings rather than the little-used "fan of" check-heart that the header of the recs section ("Fans Also Like") suggests.

My most-played game, Magnate, has only 28 fans, but 193 ratings and 104 comments. Its recommendations list consists mainly of other Decktet games and Icehouse games, and is notably devoid of other Catan-like resource management games. A rare indie like KIKA has 4 fans, 4 ratings, and 4 comments, and no recommendations, while a big hit like Terraforming Mars had 2.7K fans, 35K ratings, and 5.3K comments, with an interesting list of recommendations that included lighter games than I was expecting from it, but also reflected the strange time bias frequently noted in the announcement thread. (Readers of the thread were also quick to discover a weird and buggy recommendation of a pessimal old game on Blooms.)

Also noted in the thread was another, well-hidden method of accessing game lists by shared ratings: Other Rated Games by Users that have rated Magnate between 9 and 10. (You will have to edit the URL manually to change the game, but other variables are adjustable on the form.)

There ought to be a graph, though unlike the correlation tool, none of these rec tools provide one. I was curious about what would have happened if the recs really were based on fans, so I tried to plot some fans.

Edit: the dataset I used didn't include fans-of; the column I thought was fans was actually "votes" for something else. Also, it only included the top 5000 games (and is six months out of date), so I won't be able to see the odd fan behavior I expect on obscure games without a longer foray into the API.

Edit II: I've counted up the fans myself (details to follow in a future post) in order to create a graph of fan counts vs. ratings for the top 5000 games.

Edit III:: I've colorized the original image:

From gallery of fiddly_bits

The outliers are an interesting bunch:

name fans year rank
Dominion 5863 2008 75
Pandemic 5525 2008 74
Agricola 5394 2007 24
Carcassonne 4900 2000 144
Arkham Horror 4766 2005 258
7 Wonders 4694 2010 45
Catan 4648 1995 312
Gloomhaven 4410 2017 1
Scythe 3963 2016 7
Android: Netrunner 3723 2012 43
Twilight Struggle 3635 2005 5
Mage Knight Board Game 3479 2011 25
Small World 3390 2009 201
Puerto Rico 3379 2002 17
Star Wars: X-Wing Miniatures Game 3249 2012 71
Battlestar Galactica 2975 2008 65
Race for the Galaxy 2959 2007 48
The Lord of the Rings: The Card Game 2889 2011 108
Eldritch Horror 2846 2013 55
Ticket to Ride 2786 2004 135
Descent: Journeys in the Dark (2nd Ed.) 2773 2012 81
Lords of Waterdeep 2752 2012 53
Eclipse 2714 2011 38
Terraforming Mars 2713 2016 4
The 7th Continent 2704 2017 15
Power Grid 2692 2004 36
Terra Mystica 2662 2012 10
Dead of Winter: A Crossroads Game 2570 2014 73
Robinson Crusoe 2552 2012 41
King of Tokyo 2508 2011 251

I tossed in the year as a sign of my suspicions. More graphs of fans to follow...
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Wed Jan 30, 2019 6:14 pm
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Correlation Causation

M. C. DeMarco
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Long ago, BGG had a user-to-user correlation tool. So long ago, in fact, that user
Mikko Saari
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http://www.lautapeliopas.fi/ - the best Finnish board game resource!
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wrote a replacement for it back in 2006. A few bugs were knocked out of it early on and features were added over the years, but in my recent experience it's still a bit flaky: it doesn't always respond, and the error message you get does not always reflect the actual problem. More problematically, it is very demanding in terms of numbers of games you've rated, and it doesn't use the most appropriate correlation coefficient. [Corrected link.]

So, when I compared myself to all BGG users, even at the minimum threshold of 50 games, I only got the BGG average rating compared to my 112 games rated at the time. Those results looked something like this:

User: BoardGameGeek average rating
Corr: 0.52
#: 112

Despite the lack of compatible users, I managed to compare my ratings to those of the incomparable
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I have preserved the results here for others like me who don't rate much and may have trouble finding anyone on BGG who has rated the same 50 games as themselves.

First, the summary:

Correlation: 0.27

Average for fiddly_bits: 7.15
Average for russ: 6.63

Next, the pretty picture, captioned "The graph shows your ratings along the bottom and the other person's ratings along the side. The darker the dot, the more games fall in that spot."

From gallery of fiddly_bits

Finally, the source data: games both I and Russ rated, along with the (rounded, usually up) difference in our ratings. You can sort by difference (in theory, by clicking on the word Diff, but in practice I only got it working by adding &sort=diff to the url).

From gallery of fiddly_bits

(I saved this as an image because the margin was too small to reformat the data.)

I rated a bunch more games (I didn't start out with 112) hoping to get better data for this post, but Russ is still my only known game soulmate. If I were feeling ambitious, I would search for more soulmates by writing a tool that used a better correlation statistic.
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Wed Jan 23, 2019 7:51 pm
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The Year in Plays (2018)

M. C. DeMarco
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This inaugural blog post was inspired by a similar one by P.D. Magnus, Much played games of 2018, who also made use of the GameChat League Games Played Formatter, and in particular the option to rank by play count. I amalgamated my results a bit (manually) because I logged some plays under their expansions, variants, or reimplementations, affecting the ranking. I also noticed the tool fetches incomplete plays (e.g., Dectana), but I haven't removed those. The tool's all-time values are low if I played the game before I started logging plays consistently (e.g., The Blizzard of '78 Game, mislabeled '77 because the tool doesn't pick up your chosen variants).

Pre-amalgamation, the tool reported 342 total plays from 99 different games (91 after amalgamation to the base game). The theme of this new blog is graphs, so here's a basic barplot of the full dataset I tossed off in R:

From gallery of fiddly_bits

Here are the data for games I played 5 times or more:

Magnate x40 (43 all-time)
Unpublished Prototype x33 (42 all-time)
Aucteraden x24 NEW!
Sprawlopolis x21 NEW!
Dominion x12 (19 all-time) + Dominion (Second Edition) x3 NEW! + Dominion: Intrigue x3 (4 all-time) + Dominion: Alchemy NEW! = x19
Splendor x6 (38 all-time) + Splendor: Cities of Splendor x9 (10 all-time) = x15
Century: Spice Road x9 NEW! + Century: Golem Edition NEW! = x10
Color Wheel x8 NEW!
Lords of Scotland x8 NEW!
Nefarious x8 (10 all-time)
TransAmerica x7 (11 all-time)
Century: Eastern Wonders x6 NEW!
Trick of the Rails x6 (9 all-time)
Window x6 NEW!
Istanbul x4 (11 all-time) + Istanbul: Mocha & Baksheesh NEW! = x5

The winner is Magnate, an old 2-player game for the Decktet which was new to me in December 2017 and which I have been playing regularly ever since. It seems to have completely displaced my previous 2-player favorite, Schotten Totten (24 plays in 2017).

The apparent runner-up, "Unpublished Prototype", includes a few one-off games from the Boston Festival of Independent Games, but is mainly made up of my playtests of a Kingdomino-style game for the Decktet. I wouldn't say it was a bad idea, but it really needs colored Decktet domino cards to work well and I haven't gotten around to printing up a set yet.

The bronze goes to a Decktet solitaire game, Aucteraden, which I also implemented online this year. Many of my other plays are of Decktet solitaire games (Window, Jacynth, Quincunx, plus a zillion plays of Myrmex which I don't bother to log) or other solitaires (Sprawlopolis, Color Wheel).

Otherwise, the top of the list includes mostly nouveau classics (Dominion, Istanbul) and the relative hotness (Century, the Splendor expansions), plus a few games that either are new and exciting to me (Lords of Scotland, Trick of the Rails, KIKA), or got played mainly for supporting high player counts (TransAmerica, Nefarious).

Because the graph was looking heavily weighted towards solitaires, I chopped them out as well as I could (some games can be either, so there's still, for example, one Sprawlopolis cooperative play in the data). I removed one incomplete game as well, bringing the count down to 241 plays of 83 games:

From gallery of fiddly_bits

It really shows off my Magnate obsession vs. the long tail of the cult-of-the-new. (I didn't have any single plays of a solitaire game to remove.)

About the data: The tool I used was intended to output BGG forum formatting, but I've been doing a lot of scraping of game data (elsewhere) recently so that seemed like a minor cleanup. The problem with expansion play logging is a bit more difficult to deal with because users handle this differently. On an individual player basis you can just decide how to count expansion plays, but in, say, a project to determine cult-of-the-new statistics for the whole BGG database, you would need special logic just to lump them together the way I did for my stats.

The tool also optionally outputs images; squares are the prettiest, though not the default:

Board Game: Magnate
Board Game: Unpublished Prototype
Board Game: Aucteraden
Board Game: Sprawlopolis
Board Game: Dominion
Board Game: Century: Spice Road
Board Game: Splendor: Cities of Splendor
Board Game: Color Wheel
Board Game: KIKA
Board Game: Lords of Scotland
Board Game: Nefarious
Board Game: TransAmerica
Board Game: Century: Eastern Wonders
Board Game: Splendor
Board Game: Trick of the Rails
Board Game: Window
Board Game: Haggis
Board Game: Istanbul
Board Game: Rummikub
Board Game: Arboretum
Board Game: Dominion (Second Edition)
Board Game: Dominion: Intrigue
Board Game: GreenGreenerGreenest
Board Game: Horrible Hex
Board Game: Insider
Board Game: Isle of Skye: From Chieftain to King
Board Game: Jacynth
Board Game: Kingdom Builder: Big Box
Board Game: Oh Quay
Board Game: Paperback
Board Game: Pigment
Board Game: Quincunx
Board Game: UNO
Board Game: 27
Board Game: 5-Minute Dungeon
Board Game: Circle the Wagons
Board Game: Hansa Teutonica
Board Game: Jump Drive
Board Game: Kingdomino
Board Game: San Juan
Board Game: Sushi Go!
Board Game: Temporum
Board Game: The Great Dalmuti
Board Game: Azul
Board Game: Bananagrams
Board Game: Cover Your Assets
Board Game: Blizzard of '77 Travel Game
Board Game: Bohnanza
Board Game: Carcassonne
Board Game: Century: Golem Edition
Board Game: Citadels: The Dark City
Board Game: Clans of Caledonia
Board Game: Coloretto
Board Game: Cytosis: A Cell Biology Board Game
Board Game: Dectana
Board Game: Dominion: Alchemy
Board Game: Donner Dinner Party
Board Game: Elder Sign
Board Game: Equinox
Board Game: Exploding Kittens
Board Game: Firefly Fluxx
Board Game: Five Tribes
Board Game: Fjords
Board Game: Glen More
Board Game: Gnomi
Board Game: Guillotine
Board Game: Hero Realms
Board Game: Ingenious
Board Game: Istanbul: Mocha & Baksheesh
Board Game: Kingdom Builder: Capitol
Board Game: Le Havre: The Inland Port
Board Game: Lunch Money
Board Game: Machi Koro: Deluxe Edition
Board Game: New World Magischola House Rivalry
Board Game: Pictomania
Board Game: Potion Explosion: The Fifth Ingredient
Board Game: Pylon
Board Game: Qwirkle
Board Game: Ra
Board Game: Rumble in the House
Board Game: SET
Board Game: Sagrada
Board Game: Salem
Board Game: Scrabble
Board Game: Sleuth
Board Game: Spectral Rails
Board Game: Star Trek: Catan
Board Game: Sushi Go Party!
Board Game: Tak
Board Game: Tarot
Board Game: Terraforming Mars: Prelude
Board Game: Tesserae
Board Game: The Castles of Burgundy
Board Game: The Red Dragon Inn
Board Game: Tichu
Board Game: Ubongo
Board Game: Ultimate Tic-Tac-Toe
Board Game: Yspahan

And finally, the rest of the data, the games with fewer than 5 plays:

Haggis x4 (5 all-time)
Kingdom Builder: Big Box x3 (5 all-time) + Kingdom Builder: Capitol NEW! = x4
Rummikub x4 (5 all-time)
Arboretum x3 NEW!
GreenGreenerGreenest x3 NEW!
Horrible Hex x3 NEW!
Insider x3 NEW!
Isle of Skye: From Chieftain to King x3 (9 all-time)
Jacynth x3 NEW!
Oh Quay x3 NEW!
Paperback x3 (5 all-time)
Pigment x3 NEW!
Quincunx x3 NEW!
Sushi Go! x2 (4 all-time) + Sushi Go Party! (5 all-time) = x3
27 x2 NEW!
5-Minute Dungeon x2 NEW!
Circle the Wagons x2 NEW!
Hansa Teutonica x2 (9 all-time)
Jump Drive x2 (3 all-time)
Kingdomino x2 NEW!
Mission to Planet Hexx! x2 NEW!
San Juan x2 (7 all-time)
Temporum x2 (3 all-time)
The Great Dalmuti x2 (16 all-time)
Azul (2 all-time)
Bananagrams (2 all-time)
Big Deal NEW!
Blizzard of '77 Travel Game NEW!
Bohnanza (3 all-time)
Carcassonne (2 all-time)
Citadels: The Dark City NEW!
Clans of Caledonia (6 all-time)
Coloretto (2 all-time)
Cytosis: A Cell Biology Board Game NEW!
Dectana NEW!
Donner Dinner Party NEW!
Elder Sign NEW!
Equinox NEW!
Exploding Kittens (2 all-time)
Firefly Fluxx NEW!
Five Tribes NEW!
Fjords (2 all-time)
Glen More (2 all-time)
Gnomi (8 all-time)
Guillotine (3 all-time)
Hero Realms NEW!
Ingenious (3 all-time)
Le Havre: The Inland Port NEW!
Lunch Money NEW!
Machi Koro: Deluxe Edition NEW!
New World Magischola House Rivalry NEW!
Pictomania NEW!
Potion Explosion: The Fifth Ingredient NEW!
Pylon NEW!
Qwirkle (2 all-time)
Rumble in the House (2 all-time)
SET (2 all-time)
Sagrada NEW!
Salem NEW!
Scrabble (2 all-time)
Sleuth (12 all-time)
Spectral Rails NEW!
Star Trek: Catan NEW!
Tak NEW!
Tarot NEW!
Terraforming Mars: Prelude NEW!
Tesserae (2 all-time)
The Castles of Burgundy NEW!
The Red Dragon Inn NEW!
Tichu (4 all-time)
Ubongo NEW!
Ultimate Tic-Tac-Toe NEW!
Yspahan NEW!
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Wed Jan 2, 2019 6:08 pm
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