Nightmare Solitaire

This Riddler problem is a probability question about an old favorite: Solitaire!

While killing some time at your desk one afternoon, you fire up a new game of Solitaire (also called Klondike, and specifically the version where you deal out three cards from the deck at a time). But your boredom quickly turns to rage because your game is unplayable — you can flip through your deck, but you never have any legal moves! What are the odds?

Here is my solution:
[Show Solution]

One thought on “Nightmare Solitaire”

  1. Hi Laurent,
    Very nice. I found the same result, but I left a lot more of the work to the computer…
    https://theorie.ikp.physik.tu-darmstadt.de/qcd/solitaire.c
    I searched explicitly for all choices of 7 “face” cards which can’t be placed on each other,
    and for each I kept track of how many cards were excluded for the 8 cards which come next.
    I find the exact same rational expression.
    But I have to say, your approach got a lot closer to the answer analytically before one finally (presumably) uses a computer to do the final sums.

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