I would like to know what fraction of Spider Solitaire games (played with 104 cards and four suits) are provably 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. By "provably unwinnable" I mean that it can be demonstrated from logical argument that a game, or a class of games, is fundamentally unwinnable.
I am (so far) aware of two classes of what I currently believe to be provably unwinnable games. The wording of this last sentence was carefully chosen to convey the error bars that I assign to my present understanding.
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. 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. An example will help to make this clear. 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
Row 2 >> X X X X X X X X X X
Row 3 >> X X X X X X X X X X
Row 4 >> X X X X X X X X X X
Row 5 >> X X X X X X X X X X
Row 6 >> AS AS 3S 3S 5S 5S 7S 7S 9S 9S
Row 7 >> AH AH 3H 3H 5H 5H 7H 7H 9H 9H
Row 8 >> AC AC 3C 3C 5C 5C 7C 7C 9C 9C
Row 9 >> AD AD 3D 3D 5D 5D 7D 7D 9D 9D
Row 10 >> 2S 2S 4S 4S 6S 6S 8S 8S 10S 10S
Row 11 >> 2H 2H 4H 4H 6H 6H 8H 8H 10H 10H
Such a game has no possible moves and is therefore provably unwinnable.
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. I call this wall the Wall of Death. An example will be of benefit. 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
Row 2 >> Y QS Y Y Y Y 4C Y 4D Y
Row 3 >> 4C Y Y Y QD Y Y QS QH Y
Row 4 >> Y 4D Y Y Y Y 4S Y Y QD
Row 5 >> QC QH 4S Y Y Y 4H Y 4H Y
Row 6 >> JS JS 3S JD 3C 3H JC 3D 3S JH
Row 7 >> Y Y Y Y Y Y Y Y Y Y
Row 8 >> Y Y Y Y Y Y Y Y Y Y
Row 9 >> Y Y Y Y Y Y Y Y Y Y
Row 10 >> Y Y Y Y Y Y Y Y Y Y
Row 11 >> Y Y Y Y Y Y Y Y Y Y
Row 6, composed solely of Jacks and 3's, is 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. Similarly, none of the 3's can ever be moved because all eight 4's are buried behind the Wall of Death. In such a game all of the cards that comprise the Wall of Death can never be moved. Such a game is provably unwinnable. 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. 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
Row 2 >> Y QS Y JD Y Y 4C Y 4D Y
Row 3 >> 4C Y Y Y QD Y Y QS QH Y
Row 4 >> Y 4D Y Y Y Y 4S Y Y QD
Row 5 >> QC QH 4S Y Y Y 4H Y 4H Y
Row 6 >> JS JS 3S Y 3C 3H JC 3D 3S JH
Row 7 >> Y Y Y Y Y Y Y Y Y Y
Row 8 >> Y Y Y Y Y Y Y Y Y Y
Row 9 >> Y Y Y Y Y Y Y Y Y Y
Row 10 >> Y Y Y Y Y Y Y Y Y Y
Row 11 >> Y Y Y Y Y Y Y Y Y Y
and the game remains unwinnable.
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. An example will help to make this clear.
Row 1 >> Y QC 3S Y
Row 2 >> Y QS Y Y Y Y 4C Y 4D Y
Row 3 >> 4C Y Y Y QD Y Y QS QH Y
Row 4 >> Y 4D Y Y Y 3S 4S Y Y QD
Row 5 >> QC QH 4S Y Y Y 4H Y 4H Y
Row 6 >> JS KS 3S JD 3C 3H JC 3D 3S JH
Row 7 >> Y Y Y Y Y Y Y Y Y Y
Row 8 >> Y Y Y Y Y Y Y Y Y Y
Row 9 >> Y Y Y Y Y Y Y Y Y Y
Row 10 >> Y Y Y Y Y Y Y Y Y Y
Row 11 >> Y Y Y Y Y Y Y Y Y Y
In this example I have inserted the King-of-Spades into row 6, column 2. But note that I have also buried both of the Three-of-Spades behind the wall. 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. 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.
Note also that a game may belong to both the No Possible Move class and the Wall of Death class.
At this juncture I think it worthwhile to ask questions that are somewhat more refined than my original question.
Q1: What fraction of games are in the No Possible Move class of games?
Q2: What fraction of games are in the Wall of Death class of games?
Q3: What fraction of games are in both the No Possible Move class and the Wall of Death class of games?
Q4: Are there other classes of provably unwinnable games, and if so, what fraction of games are in those classes?
Q5: Are there games that are unwinnable but not provably unwinnable? This needs some elaboration. Recall that my definition of provably unwinnable was that a game can be demonstrated unwinnable by means of logical argument. This question concerns whether there exist truly unwinnable games that cannot be proven to be unwinnable by means of logic. Such games could in principle be proven unwinnable by a brute-force search of the entire tree of possible moves.
I am interested in both analytic and numerical-simulation answers to these questions. I have made a very crude estimate of the answer to Q1. I estimate that roughly $1$ in $10^{13}$ games (give or take an order of magnitude) are in the No Possible Move class.
I suspect (but do not know) that one could play 10 games of Spider per day for their entire life and never come upon a provably unwinnable game. The implication (of course) is that when you loose a game it is almost certainly because you gave up, not because the game was unwinnable.
Note Added: A Spider-playing computer program (plspider) is reported to have a win rate of 99.994%, loosing just 2 out of 32,000 unique games that were tested. This lends credence to the notion that provably unwinnable games are extremely rare. Of some interest is game #14934 - a game that the website author(s) says "appears to be unwinnable." Upon examination it can be readily discerned that game #14934 is neither a No Possible Move game or a Wall of Death game. If it is truly unwinnable then either: 1) it is a member of a new class of provably unwinnable games, or 2) it proves the existence of unwinnable games that are not provably unwinnable games (except by brute-force searching of the entire tree of moves). Kudos to the author(s) of the website for their care in stating that the game "appears" to be unwinnable.