Hi,

We are working on this for use as a potential puzzler on the NPR
program "Car Talk." If anyone can show us some calculations for this,
we'll be eternally grateful.

You are playing a standard game of solitaire, in which you begin with
7 piles of cards. After you turn over the top card in each pile you
see three 5's.  What are your odds of winning the game from this
starting position?  Do the odds change if you turn over three 10's? Or
three 3's?

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Clarification of Question by cartalk-ga on 11 Mar 2005 08:30 PST

To clarify:


How do the odds of seeing 3 of a kind compare to 2 of a kind, to 0 of a kind?

Not to great on statistics at the moment but here's my logic on it.

The higher the number, the more time it takes to get rid of it in the
game. So the odds of starting with 333 are better than 555, and odds
at 555 are better than 101010.

The words odds is ambiguous as people can choose diffent cards. If you
mean a perfect game, then the odd would be a hell of a mathematical
algorithm to compute still. Not quite a chess match but not quite a
checkers picnic either.

Starting with three of a kind is not too unlikely. The only way I can
see it impacting any is if the number is fairly low. Perhaps less than
4? That would likely be a calculus problem to figure out the point of
inflection.

This is too complicated... even writing a brute force algorithm would
be hard since you have to get it to solve each possibility.  My
solution... pick a different problem

Where is mathwrite-ga?
I don?t think an absolute odds on any starting game line-up is
possible, but do think that three-of-a-kind lowers the odds because it
will reduce by one the number a cards you can play on (having two red
fives) and the chances of being able to shift them (having to find
both black sixes).
But I no really nothing about statistics and this may very well not be true.
I don?t believe the number makes any difference.  Yes, you may be able
to play off threes sooner, but if the aces are hiding, you?re stuck
with dead end on that stack until you have a four to shift the three
to, whereas you can play more cards onto the tens.

Shucks!  I don?t think it makes any difference with any three-?kind,
or with one at all, any card can come up at any time ?
Interesting to know what the card-stats folks say.

Even if three of a kind made a difference... it would be extremely
difficult to show...

The question has some serious problems with it.  First, what is
"standard solitaire"?  Klondike and Spider are both fairly standard
solitaire games, and there are probably others that are also
considered standard.  Klondike and Spiderette both have seven piles of
cards in a triangle.  If you mean either of these two games, then as
the other commenters said, computing exact probabilities is probably
an intractible combinatorial problem, because it depends on computing
exactly optimal play.  The exact probability not only likely depends
on which three-of-a-kind, it also likely depends on the other four
cards at the top of the piles.

Kuperberg-ga,
Let's assume/agree that Cartalk is asking about Klondike.
You have a point about the other four cards; could be that the triplet
disappears quickly:  I don't have explain to you all how.  But that
possibility  - like everything else in the game -  just depends on the
random order of the cards in the whole pack.  Anything can happen.
Very interesting is the idea of optimum play.  I do not know if it is
considered a rule of the game that one must play every card from the
talon(stack) which can be played; not doing so because it would be the
third card played and then in the next round less unseen cards would
be available.
Oro if one must play a card from the tableau, which one might not want
to do if it left a blank or one knew that a card from the stack could
be played later instead.
If the question could be solved statistically, then it would seem that
these choices would have to be eliminated to keep everything in the
realm of pure statistics.
I don't think it can be solved.

To play optimably - if there is no limit on the number of times you go
through the stack - I would think one should go through it first with
out playing a card but observing the order of them (with your
"Rainman" like mind), calculating if you can play this or that card,
what will be available the next time, and so on each time.  Of course,
this is mentally  - virtually -  impossible   and has nothing to do
with question.
Just an idea for the forum.

This sounds like an issue of P vs NP.  Deciding whether or not you
could win a certian game of solitaire where 3 of a kind are showing at
the beginning is a P type of problem, we can use a computer run
through it quickly and decide if it is soveable depending on where all
the other cards are in the deck at that time.  Finding the odds of
winning with 3 face cards up is an NP type of problem, because it
would require too many checks to find out in a real ammount of time. 
More about P vs NP problems here:
http://www.claymath.org/millennium/P_vs_NP/