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Solitaire: Probability Of No First Move (Posted on 2021-01-12) Difficulty: 3 of 5
Define a legal Solitaire-type move within a group of face-up playing cards as moving one card to partially cover a stationary card provided that the stationary card:

1) has numeric value exactly one larger than the card to be moved; and

2) is of a color opposite to that of the card to be moved.

Note that Jack, Queen and King have numeric values of 11, 12 and 13 respectively and Ace only has the value of 1.

-----

From 52 cards, 8 cards are randomly chosen and are laid out face up in a row. In this simplified game, the other 44 cards are discarded. What is the probability that no legal Solitaire-type move is possible for a:

a) Standard Deck, 2 suits are red and 2 suits are black.

b) One-Suit Deck: what if the 52 cards are all one suit, 4 complete sets of Ace through King, and the "opposite color" requirement is removed?

No Solution Yet Submitted by Larry    
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Solution simulations | Comment 1 of 5
Each running of the program below does a simulation of a million trials. Five independent run outcomes are shown:

moveCt=0; moveColorCt=0;
for trial=1:1000000
    had=zeros(1,52);
    for card=1:8
        found=false;
        while found==false
           c=randi(52);
           if had(c)==0
               found=true;
               deck(card)=c;
               had(c)=1;
           end
        end
        val=mod(c,13); 
        if val==0 val=13;
        end
        deckv(card)=val;
        deckc(card)=mod(c,2);
    end
    hadMove=false;
    hadMoveWithColor=false;
    for i=1:8
        for j=1:8
            if abs(deckv(i)-deckv(j))==1
                hadMove=true;
                if deckc(i)~=deckc(j)
                    hadMoveWithColor=true;
                end
            end
        end
    end
    if hadMove
        moveCt=moveCt+1;
        if hadMoveWithColor
            moveColorCt=moveColorCt+1;
        end
    end
end

disp([1-moveCt/trial 1-moveColorCt/trial])




Each run represents the frequency of no first move in one million trials, the first is for the color-free version; the second is with the color requirement.

>> solitaireNoFirstMove
                  0.020443                  0.102045
>> solitaireNoFirstMove
                  0.020597                  0.101702
>> solitaireNoFirstMove
                  0.020579                  0.101896
>> solitaireNoFirstMove
                  0.020586                  0.102319
>> solitaireNoFirstMove
                  0.020623                  0.102136
                  
Without the color requirement, the probability of no move is apparently between 2.0% and 2.1% but closer to the latter. With the color requirement it's about 10.2%.                  

  Posted by Charlie on 2021-01-12 10:51:48
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