![]() This question concerns whether there exist truly unwinnable games that cannot be proven to be unwinnable by means of logic. Recall that my definition of provably unwinnable was that a game can be demonstrated unwinnable by means of logical argument. Q5: Are there games that are unwinnable but not provably unwinnable? This needs some elaboration. Q4: Are there other classes of provably unwinnable games, and if so, what fraction of games are in those classes? Q3: What fraction of games are in both the No Possible Move class and the Wall of Death class of games? Q2: What fraction of games are in the Wall of Death class of games? Q1: What fraction of games are in the No Possible Move class of games? Note also that a game may belong to both the No Possible Move class and the Wall of Death class.Īt this juncture I think it worthwhile to ask questions that are somewhat more refined than my original question. So to utilize a King as part of a Wall of Death one must bury at least one pair of value-equivalent cards of the same suit as the King behind the Wall of Death. In doing so, I have created a situation in which the King-of-Spades can never be lifted because to do so would require completing a continuous series of Spades from the King-of-Spades down to the Ace-of-Spades, and such a continuous series cannot be constructed without a Three-of-Spades. ![]() But note that I have also buried both of the Three-of-Spades behind the wall. In this example I have inserted the King-of-Spades into row 6, column 2. It is also possible for a King to be part of a Wall of Death, but the nature of what cards must be buried behind the wall is somewhat different. For instance, the Jack in column 4 of the above game could be relocated from row 6 to row 2: Row 1 > Y QC Y Y Note that each column must contain a member of the Wall of Death but that the member of the Wall of Death need not be in row 6. In such a game all of the cards that comprise the Wall of Death can never be moved. Similarly, none of the 3's can ever be moved because all eight 4's are buried behind the Wall of Death. None of the Jacks can ever be moved because all eight Queens are buried face down in rows 1.5 behind the Wall of Death. Row 6, composed solely of Jacks and 3's, is the Wall of Death. Consider the following game in which rows 1.5 are dealt face down, "Y" denotes a card of arbitrary value and suit, and rows 7.11 are in the stock: Row 1 > Y QC Y Y The "Wall of Death" Class: This is a class of games in which all of the cards that would enable a lifting of a card are dealt face down and "walled off" by other cards that also cannot be lifted for the same reason. Such a game has no possible moves and is therefore provably unwinnable. Consider the following game in which the cards denoted by "X" are dealt face-down and rows 7.11 are in the stock: Row 1 > X X X X Two criteria must be satisfied for a game to fall into this class: 1) in each row of face-up dealt cards there must be no moves between the face-up dealt cards, and 2) there must be no moves that are enabled by picking up multiple cards. ![]() The "No Possible Move" Class: This is a class of games in which there are no moves at all, regardless of the skill of the player. The wording of this last sentence was carefully chosen to convey the error bars that I assign to my present understanding. I am (so far) aware of two classes of what I currently believe to be provably unwinnable games. By "provably unwinnable" I mean that it can be demonstrated from logical argument that a game, or a class of games, is fundamentally unwinnable. ![]() It seems various people have various ideas about what constitutes an unwinnable game, and so I will restrict my question to "provably unwinnable" games. I would like to know what fraction of Spider Solitaire games (played with 104 cards and four suits) are provably unwinnable.
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