37

It's not hard to prove that an unsolvable start exists. Just imagine a start where the only possible first moves would be moving cards to the extra cells. In some versions, -1 and -2 are examples of this though the only way to play them is to choose that seed. If you only count setups which can exist in normal play, seed 11982 in the Windows version is an ...


16

The hardest part of computing the probability is determining the best strategy for the players. I have used a script to simulate games where players use a few simple tactics, and estimated the probability based on these strategies. Given that the strategies I implemented are not the optimal strategy, these figures provide an upper bound on the probability ...


12

Meaningful decisions per hour is one of several contributory factors, not the only one. Other important components of "fun", at least for me, include: high ratio of meaningful decisions to total decisions Interesting choices to make in those decisions Interesting setting/backstory visual appeal link of setting/backstory/theme to mechanics. interactivity of ...


11

That's very, very simple in fact: shuffle the discard pile constantly. Take a player that, while not playing his turn, shuffles the discard pile of cards. When discarding, simply give the cards to this player (who can change when he's tired), he will put them in the deck and go on shuffling.


10

Linked in Probabilities in the Game of Monopoly® by Truman Collins, are C source code files written to calculate the probabilities of normal Monopoly. We can just roll up our sleeves and tweak the program ourselves to see what kind of changes we get. Choices There are two separate programs linked: one that calculates the probabilities based on simulated ...


9

A friendly warning: This is a looong answer. I've calculated the chances of Fools' Landing being lost immediately after the first turn with all players playing to prevent it and a random tile setup. A few variables: n = number of players w = water level after rising once (2 for Novice, 3 for everything else) Special abilities/actions First, of all,...


9

The key change in reversing the direction of movement is the fact that players now leave Jail in the opposite direction. The oranges and reds are particularly profitable because they are within 1-2 rolls of Jail, among other reasons. When moving counterclockwise, the dark purples and dark blues are now in a more often visited area, so I would expect their ...


8

Key assumption: Every player places both of their initial settlements according to what will get them the most resources (so we don't have people going for port combos, weird 12/2 superstitions, etc.) I just downloaded the rules and have the Beginner's Setup in front of me. If you don't, then this won't make any sense! We all agree that it suffices to add ...


8

councilroom.com hosts stats for millions of online games played at dominion.isotropic.org. It's currently down but it should give you the most accurate answer when it goes back up again. For example, here is a ranking of the best openings from google's cache


8

Here are the (quite extensive!) results: http://www.tkcs-collins.com/truman/monopoly/monopoly.shtml The first part of that page is the long-term probabilities for ending up on a particular square. The last column tells you the ranking: you will end up in Jail most often, Illinois second-most-often, then Go, then the B&O Railroad, etc. This is neat, but ...


8

Instead of modeling you can compute these results exactly. If you roll n dice at difficulty d, the variance will be [d * n * (d-1)/6 * (7-d)/6], and the mean net gain will be [d * n * (7-d)/6 - 1]. Both scale linearly with n, so here are the mean (measured in Net Gain Per Die) and variance for each value of d: ----------------------- | d | Mean | Variance | ...


7

Greed (a.k.a. Farkle) has an extensive amount of statistical analysis done on the game. Now that you have updated your question, here is my updated answer. If you want to maximize your score, what you need to figure out is what is your expected score if you risk rolling, compared to your score if you choose not to roll. So, the first thing you need to know,...


6

The article on Shuffling at the Wikipedia discusses the famous paper by Persi Diaconis and Dave Bayer which shows that a 52 card deck of cards doesn't become random until the fifth shuffle, and requires seven shuffles to become "truly" random. Your question "Are there any other methods of cuttings of the deck or shuffle types that create a more random deck?"...


6

The answers to your questions depend on the details of how you score the game. For example, Facebook Farkle scores three pairs as 750 points, Gaby Vanhegan's Zilch implementation scores it as 1500 points, and Wikipedia's entry on Greed scores it as 800. These and other scoring differences affect the expected value of rolls and ultimately the optimal play ...


6

Summary The probability of losing in the first turn due to Fool's Landing sinking, assuming all players try their hardest to avoid it, is dependent on the difficultly level and the number of players n: Difficultly | n | Probability Novice | 2 | 0.00199 Novice | 3 | 0.00122 Novice | 4 | 0.00071 Normal/Elite | 2 | 0.00291 Normal/Elite | 3 |...


6

I know the results. Red and Orange properties are places where players most often lands on. Most often visited places are the Jail and the 3rd red property ("Illinois Avenue" in the American version and "Avenue Henri-Martin" in the French version). I found a table here (fr). The table contains the French names so I join a french Monopoly board to this ...


5

The card that immediately pops to mind: Here's an article on dominionstrategy.com which outlines why this card is considered the best in the game. No other card allows you to shape your deck so quickly and early, ensuring you get useful and powerful draws to ramp up your deck. If you look at the best openings ranking linked by Dor Shemer, you'll see many ...


5

These were all more or less directly copied from the source attributed at the bottom of the answer: Directly rolling a particular number (e.g. 2) 30.55% Rolling a particular double (e.g. 3-3) 2.77% Rolling a particular non-double (e.g. 5-1) 5.54% Rolling any double 16.66% Chance of getting off the bar with one or two pieces and X open points: ...


5

Try this, it's a system called Whole-History Rating. From the abstract: Whole-History Rating (WHR) is a new method to estimate the time-varying strengths of players involved in paired comparisons. Like many variations of the Elo rating system, the whole-history approach is based on the dynamic Bradley-Terry model. But, instead of using incremental ...


5

I highly recommend Winning Monopoly if you are interested in such analysis. As other answers point out, payback period (combining rent amount and frequecy) is key for making winning decisions. One of the point from this book that I haven't seen else where is determining what version of Monopoly you are playing. Are you really playing until all others are ...


5

The reason that equity is used instead of winning probability is because it is possible to win a single game, a double game (gammon) or triple game (backgammon). Let's say that the value of the game, or bet, is $1. (That would occur if the cube is in the middle. If it has been turned, you multiply by 2, 4, or whatever the number is on the cube.) Let's ...


5

Accordion According to one of the biggest player on the solitaire market (SolSuite), the game of Accordion (one card at a time version) has a chance of winning in about 1 of 200 games, i.e. 0,5% But I would actually say that there is no difference in the winning percentage (if played right) between the two versions you are talking about, since (in the ...


4

(disclaimer: i haven't played button men seriously in ages, but i'll share my impressions from back in the day) My feeling is that Lab Rats is the best button, as its 2-dice will allow skill-attacks on far better dice than it has to give away. Bunnies, with its 1-dice is also strong, but can too-often get trapped into not being able to make any attacks. ...


4

With Ticket to ride, 3-5 riffle shuffles should be sufficient to randomize the deck sufficiently for play. As long as the runs of matched cards aren't 5-6 cards long, it's not a big issue. I'll note that japanese style block shuffling is inadequate for TTR until about the 9th or time through the deck, which I discovered due to using sleeved cards in Nordic....


4

I am not sure which particular site you are referencing, but How to Win at Monopoly calculated the Return On Investment (ROI) of all the properties based on the liklihood of landing on them. They concluded that Railroads (especially 4) had the fastest most consistent ROI, followed by a Monopoly with 3 Houses, utilities were the worst ROI.


4

So looking online I found this from 2015. It says More are sold per capita in Germany than anywhere else on earth Magic the Gathering is big, with something like 20 million players worldwide, however that will pale in comparison to board game sales so I'd be likely to ignore card games as they won't add a huge amount overall compared to board games. ...


3

it is also easier to go for the longest road when you start in fourth or third. keep that in mind. To me, it all depends on the board : like if there is one supreme location and players takes it... but that's kind of rare. There are usually 4 honestly good places and then it gets worse, so the advantage of being first is not that good to me.


3

I'd learn the chances for the roll combinations. There are 36 possible rolls, (let's say of one red and one green die) as follows: 6-6: 1/36 11: 2/36 (two 6-5s) 5-5: 1/36 10: 2/36 (two 6-4s) 9: 4/36 (two 6-3s, two 5-4s) 4-4: 1/36 8: 4/36 (two 6-2s, two 5-3s) 7: 6/36 (two 6-1s, two 5-2s, two 4-3s) 3-3: 1/36 6: 4/36 (two 5-1s, two 4-2s) 5: ...


3

With every card game I play I do a mix of Riffle, Stripping, and Mongean-Stripping Combo. I generally intermix these in a random order with a random number of times, usually at least 3. I've found that mixing shuffling techniques, especially in a random order, will result in a fairly nice card distribution and randomization.


3

Generally speaking, I shuffle using a combination of overhand and Hindu shuffle. I start with the pack in my left hand, pluck out the middle half with my right, and then proceed to drop blocks of cards alternatively onto the top and then bottom of the pack, using a quick flick of fingers or thumb to swap between cards dropping onto the top or bottom of the ...


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