39

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 ...


11

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 ...


11

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 ...


8

The short answer is "Yes, but...". The longer answer is, as per the paper in question, that a team of researchers did some calculations on what would happen in a 2-player game of Monopoly where both players follow very simple strategies (and a couple of things that aren't 100% by the rules), notably: Always try to keep a small reserve of cash on hand to ...


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

According to the accepted answer on the post containing that number, it originates in the book How to Win at Gin Rummy: Playing for Fun and Profit by Pramod Shankar. Unfortunately, the book doesn't seem to have a source or proof for it. Thankfully, though, a comment on that post does give an answer. A blog called Entropy Clay steps through the calculations, ...


6

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

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 ...


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

My original answer assumed that there was no penalty for guessing incorrectly. This answer assumes that you are playing with the rule that incorrect guesses cause a game loss. My original answer is provided below this one Never guess, unless a player has only a single character left If your opponent guesses, then your odds of winning are 50% if they have ...


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

This question's been unanswered for a while, so I thought I'd take a crack at it. First up, I should say that I haven't looked at the four player case, just the two player game. The main reason for this is that it makes presenting the data a lot easier. I've put some graphs later, and you can see that they're already hard enough to understand for the two ...


3

Playing online against 3 computer opponents, one of my games would typically last 30-45 turns. I used the standard rules (i.e., no money on Free Parking, limited number of houses and hotels). Because it was me against computers, the games would take ~15 minutes to play. Playing against human opponents would certainly extend the actual playing time, and may ...


3

For me, I would divide the deck into smaller parts that you can handle. Riffle shuffle (or any way you like) those and mix the small decks together, one group at a time.


2

The chance of getting a total of x on the two dice is simply (6-|7-x|)/36 where |7-x| is the absolute value of 7-x.


2

You can "translate" the word "equity" as a value of the particular position. Lets imagine we are one roll away from ending the match and we only have two checkers on deuce point. We will win with the probabilty of 26/36 and we will lose with the probability of 10/36. Lets also imagine that there is a friend who offers us some money and asks us to abandon ...


2

I would suggest checking out the Board Game Geek Avalon community. People regularly record their play sessions and the users on the forums are usually willing to go into in-depth analysis and discussion on game setup and configuration. For instance, even one of the most recent forum discussions is about statistical anomalies in Avalon games.


2

The answer to this question will be very heavily dependent on your group, and their groupthink. In my experience, often bidding 3 on the blade would not result in getting it (I would probably be 2nd or 3rd, with someoen else going all in on the sword), while a bid of 2 on the Raven would probably be enough to get me at least one star (with someone else ...


2

You would usually not skip the Go square (and the chance to collect $200) before going to Jail. The exceptions would be the Chance square between Oriental and Vermont (light blues), and the community chest square between the two dark purples. Even so, you would collect your $200 on the tenth square after leaving Jail instead of the 30th, and you would not ...


2

I haven't heard of this variation. Nuclear Wang's analysis is correct. Note the Dark Blues getting a frequency boost along with their high rent would cause ownership of that color group be key to winning a reverse game. To take the analysis further, take a look at the following site: http://www.tkcs-collins.com/truman/monopoly/monopoly.shtml This site ...


2

Three Independent sources counted about 136,694 Gin hands, out of 15.8B possible 10-cards hands. 2+2 forum post: 1 in ~118,000. Using brute force which checked for Gin each of the 15.8B hands. Rulemonger's analysis: 136,694 in 15,820,024,220 or 1 in ~115733 How to Win at Gin Rummy: 1 in ~117,000, according to the book How to Win at Gin Rummy: Playing for ...


2

Someone one the FB page where this question was originally posted found this answer from the School of Operations Research and Information Engineering Cornell University Ithaca NY 14853, USA ESTIMATING THE PROBABILITY THAT THE GAME OF MONOPOLY NEVER ENDS At the end of the 10-page report, the following is stated : All four of our estimators yield ...


2

Wizards of the Coast deliberately obscures this data. Several years ago, some fans created a website that used bots to automatically crawl Magic the Gathering Online's matches to construct a table containing the meta percentages and the win/loss rates of each deck. However, Wizards later changed how MtGO's systems worked with the specific goal of preventing ...


2

I can think of two mathematical measures that might be useful, both based around Elo ratings. Is the correct Elo distribution function. Let's suppose Player B beats Player A 64% of the time, and Player C beats Player B 64% of the time. How often does Player C beat Player A? Generally, speaking, the higher the number, the less of a role luck plays, though ...


1

Winrate when going first and second: this is clearly going to depend on the format, and since all the commonly-played formats are changing over time, this data will never remain up-to-date. Wizards does not publish the data, so it's up to players to find as much information as they can. There is nonetheless some data available: Play or draw? The Trouble ...


1

I have played Gin often with my family, at least several hundred rounds. I do not recall ever having participated in a game which ended in a draw. I will note that I have primarily participated in two-player games, logic indicates that games with additional players would have a higher probability of ending in a draw as there would be fewer rounds per player ...


1

EDIT: this count is flawed, it doesn't recognize sets of three that include a spade. The correct number is given in the accepted answer. When the code is corrected, it gives the 136,694 unique gin hands, matching the accepted answer. Original: 1 in 308,984, according to Andrew Inwood's analysis, which includes source code.


1

According to a Ted-Ed article entitled Here's how to win at Monopoly, according to math experts ... the average game of Monopoly takes about 30 turns per competitor... Reference https://blog.ed.ted.com/2017/12/01/heres-how-to-win-at-monopoly-according-to-math-experts/ So the answer to your question is 120 turns (4 players times 30 turns/player) By ...


1

In The Backgammon Book, World Champions Oswald Jacoby and John Crawford recommend with both 6-2 and 6-3 moving one runner to the bar point and one man from the 12-point into your own outer table; though they concede that there another move almost as good: The modern play [written in 1970] is to use the six to move one back man to the black bar point and ...


1

Equity is especially useful because it pre-calculates some of the analysis used in utilizing the Doubling Cube. Consider the situation described by @Skytten: We have two chequers on the 2-point to roll; opponent has two chequers on the 1- and 2-points (a guaranteed win if he/she gets to roll). Our equity is $44.44. If we offer a double to opponent our ...


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