In Klondike solitaire with standard rules (Draw 3 cards, Re-Deal infinite), a 'null' hand is a hand where you cannot play any card.
Statistically, is it more common to win the game, or have a null game?
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1Are you saying that a "null game" is the same as losing a game, since you can't play any cards and are left with only "null hands"?– ThunderforgeCommented Sep 4, 2016 at 5:06
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See also math.stackexchange.com/questions/121305/….– CohensiusCommented Jun 26, 2020 at 20:20
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6Does this answer your question? Can every game of Klondike-Solitaire be solved?– CohensiusCommented Jun 26, 2020 at 20:40
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1@Cohensius This does not appear to be a duplicate of that question as it is asking if it is possible to win every game while this one is asking what is the chance of not having a valid play at all.– Joe WCommented Jun 26, 2020 at 22:43
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1@JoeW right, both of those questions are close not the same, retract my vote.– CohensiusCommented Jun 28, 2020 at 7:01
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1 Answer
In Klondike Solitaire there are 7 cards face up on the table, 21 face down on the table, leaving (out of the 52 card pack) 24 cards in the deck. If you're dealing 3 cards at a time, only 24/3 = 8 of these cards are available. So only 15 cards are available at the start of the game.
In order for there not to be any valid moves at the start, you would need:
- No aces
- No adjacently numbered cards
This is easily achievable with 15 cards. For example:
- 4 Kings
- 4 Jacks
- 4 Nines
- 3 Sevens
According to Wikipedia, which answers the probabilities:
- About 79% of the games are theoretically winnable
- The number of games a player can probabilistically expect to win is at least 43%
- In addition, some games are "unplayable" in which no cards can be moved to the foundations even at the start of the game; these occur in only 0.025% of hands dealt
You are therefore far more likely to win the game (even if your decisions are far from perfect) than get an unplayable/"null" hand.
