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Conference rusure::math

Title:Mathematics at DEC
Moderator:RUSURE::EDP
Created:Mon Feb 03 1986
Last Modified:Fri Jun 06 1997
Last Successful Update:Fri Jun 06 1997
Number of topics:2083
Total number of notes:14613

643.0. "Las Vegas Solitaire" by WKRP::KIER (Mike DTN 432-6286 @CYO ) Fri Jan 09 1987 17:29

A friend of mine many years ago introduced me to a solitaire game he
called Las Vegas Solitaire.  I have no idea if it is actually played in
casinos but its given me many hours of enjoyment (mental gambling).  I
thought I'd introduce it here and let the mathematicians play around
with the odds of winning for the house and player if there is interest. 

The rules of play are listed below after the <ff>.  They are very 
similar to most forms of solitaire that I've seen.  The gambling rules 
are as follows:

The player "purchases" the right to play for $50.  If he doesn't 
forfeit, the house pays the player $5 for each card successfully placed 
to the Ace Board on completion of the game.

The player can maximally lose $50 (no cards to the board).
The player can maximally win $210 (all cards played to Ace Board).

Given a random shuffle and no forfeiture due to play mistakes (an 
artificial house monetary advantage), what are the odds of the player 
coming out ahead (or behind)?  By what monetary amount?

Enjoy,

	Mike
  CincinWKRPnati


Las Vegas Solitaire - Rules of Play

Deal one card face up and six cards face down, each separately to the
right of the preceding card.  Then deal one card face up on top of the
card to the right of the first card and a face down card on top of the
remaining cards to the right.  Continue this procedure until you have
dealt a face up card on every stack.  The space above this board will be
used for accumulating an ascending stack of cards by suit beginning with
the Ace of each suit; no gaps are permitted in the sequence.  Place the
remaining 29 cards face down below with a space reserved for a face up
pile to the left.  The board should look as follows: 


          Reserved       Reserved       Reserved       Reserved
            for            for            for            for
           Heart          Spade          Diamond         Club
            Ace            Ace            Ace            Ace



 ____   ____      ____      ____      ____      ____      ____
'    ` '    `_   '    `_   '    `_   '    `_   '    `_   '    `_
|Face| |Face| `  |Face| `  |Face| `  |Face| `  |Face| `  |Face| `
| Up | | Up |1|  | Up |2|  | Up |3|  | Up |4|  | Up |5|  | Up |6|
|    | |    | |  |    | |  |    | |  |    | |  |    | |  |    | |
`----' `----' |  `----' |  `----' |  `----' |  `----' |  `----' |
         `----'    `----'    `----'    `----'    `----'    `----'


          Reserved          ----
            for            '    `_
           Face            |Face| `
            Up             |Down|2|
           Stack           |    |8|
                           `----' |
                             `____'

Play *must* proceed from top to bottom and from left to right at all
times, else player forfeits.  A card plays to the Ace board if, and only
if, it is a) an ace, or b) played on the card of the same suit
immediately lower than itself in sequence (e.g. two of hearts plays on
ace of hearts).  A card plays on the main board if a) it is a king and
is played on any empty file, b) it is played on a card of opposite color
to its suit which is immediately higher than itself in sequence (e.g. a
ten of diamonds plays on top of a jack of clubs) AND it is either the
bottom most card in a file (in which case the entire face up file that
covers it moves with it), it is an uncovered face down card on the main
board and is to be turned face up, or it is the top most card in the
Face Up stack.  When there are no further plays from the main board or
the Face Up stack, the next card from the Face Down stack is turned over
and laid on the Face Up stack and scanning for potential plays resumes. 
After the last card from the Face Down stack has been turned over and
all possible plays made the game is over and the Ace board is tallied. 

So the sequence is:

1) Scan the main board from left to right and play any card to the ace 
baord that is playable; iterate until no more then proceed to step 2.

2) Scan the main board from left to right and make any play on the main 
board that is possible within the main board; if any play is made 
then return to step 1 else proceed to step 3.

3) Determine if the top card in the Face Up stack can play to the Ace 
board; if so play it and return to step 1; else determine if the top 
card in the Face Up stack can play to the main board; if so play it and 
return to step 2; else proceed to step 4.

4) turn the next card from the Face Down card over and lay upon the Face 
Up stack and return to step 3.  If there were no more cards then tally the 
ace board, settle accounts, and stop.
T.RTitleUserPersonal
Name
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643.1Klondike for kashSQM::HALLYBAre all the good ones taken?Fri Jan 09 1987 19:2310
    A friend of mine used to go to Reno and play this game.  Just a
    slight correction on the cash end:  the deck actually cost $52
    ("$1 per card") and you didn't get paid $5 for a lone Ace on the
    Ace board, you got 0.  ($10 for A-2, $15 for A-2-3, etc.)
    
    In the GAMES notesfile (KP7, etc.), note 414 describes a rather
    nice VAXstation solitaire game.  Maybe somebody can con Ray out of 
    the card-playing side of his code.
    
      John
643.2Stick to craps or blackjackMODEL::YARBROUGHMon Jan 12 1987 12:335
I doubt that anyone in Vegas has actually worked out the odds, but I 
believe they are SUBSTANTIALLY in favor of the house. It may be one of the 
worst bets (from the bettor's point of view) in Vegas. I suspect your 
expected return on a $50 investment is about $20. This is based on a lot of 
spare time spent playing the silly game.
643.3I only use mental moneyWKRP::KIERMike DTN 432-6286 @CYO Mon Jan 12 1987 13:4220
Re: .2

I'd come to pretty much the same conclusion, Lynn, based on the same 
method... a sort of statistical approximation.  I couldn't begin to 
think of a method to approach it using probabilities, so I tossed it in 
here to see if it would be taken up as a challenge by some of the more 
advanced math/probability experts.

Re: .1

Yeah, I knew it was $52 and not $50, but I tend to round it off to make 
it easier to quickly mentally tally my winnings or losses.  It shouldn't 
have a large effect on the odds.

I *didn't*, however, know about the loss of return on bare aces on the 
Ace board.  That makes a big difference and tilts the already 
unfavorable odds further in the House's direction.

	Mike
  CincinWKRPnati
643.4Skill or Chance?CHOVAX::YOUNGBack from the Shadows Again,Tue Jan 13 1987 06:169
    Its unclear to me, (I have not actually tried to play a game out)
    whether there is any 'skill' in this game or not.  
    
    In other words, is there always only 1 legal next move?
    
    If not, then it may be that the unfavorable odds mentioned earlier
    are merely a result of inferior play?  No insult intended. ( 8^) )
    
    --  Barry
643.5what are the accurate rules for solitaire, anyway?VIDEO::OSMANand silos to fill before I feep, and silos to fill before I feepTue Jan 13 1987 13:5511
I notice some slight variations in your description than what I'm used
to, and I only mention them in case you didn't intend to be as restrictive
as you were.

According to your rules, if a 5D appeared on a file, and 4D was already
in the Aces pile, you'd be FORCED to play your 5D on the 4D.  I recall
from my old rules that I had the option of leaving the 5D on its file,
in order that I can play a black 4 file on top of it to reveal more
cards.  Did you intentionally disallow this ?

/Eric
643.6DeterministicWKRP::KIERMike DTN 432-6286 @CYO Tue Jan 13 1987 15:0313
Re: .4 and .5

Yes, it is a rigid definition.  With the exception of mistakes resulting 
in forfeiture (such as playing a card out of sequence) the play of the 
game is fixed when shuffling is completed.  It could conceivably be 
represented as a relatively straight forward finite state machine.  The 
outcome of the game is solely determined by the random position of the 
cards in the deck at the time of the deal and not by any skill of the 
player.  This is where it differs from the various forms of recreational 
solitaire where plays such as Eric mentions may affect the output of the 
game.

	Mike
643.7Save those winning shuffles!SQM::HALLYBAre all the good ones taken?Tue Jan 13 1987 15:336
    It would be interesting to look at the games WON and see if there
    is anything special about the arrangements of the cards in the deck
    prior to the play.  Are there any patterns of cards that lend
    themselves to a statistically higher chance of winning?
    
    Great research for a college project.
643.8.7 doesn't make sense. shuffle is WIN or LOSEVIDEO::OSMANand silos to fill before I feep, and silos to fill before I feepTue Jan 13 1987 18:5811
Re .7:

I don't understand what you mean "patterns that have more chance of
winning".

If the rules are indeed strict, there's only one choice per move,
and no skill.

Hence a given shuffle either has a 100% or a 0% chance of being a winner.

/Eric
643.9Just a small matter of counting...WKRP::KIERMike DTN 432-6286 @CYO Tue Jan 13 1987 23:115
Ah, I think you've pointed the path to the solution, Eric.  One merely 
has to figure out how many shuffles lead to a payout of $60 or more out 
of the total possible shuffles.  I can't wait to see your answer. :-)

	Mike
643.10CLT::GILBERTeager like a childWed Jan 14 1987 03:002
    A good approximation of the odds can be gotten by applying the
    (ahem) Monte Carlo method.
643.11SQM::HALLYBAre all the good ones taken?Thu Jan 15 1987 17:0020
.8> I don't understand what you mean "patterns that have more chance of
.8> winning".
.8>
.8> If the rules are indeed strict, there's only one choice per move,
.8> and no skill.

    Right.  You got it.  ("Got what?").  Well, after shuffling you have
    a sequence of cards.  This sequence is then mapped to "WIN" or "LOSE"
    depending upon the play of the cards.  This is a well-defined mapping
    because the rules are strict.
    
    Play a million games, remembering all the sequences that map to "WIN".
    Look at the sequences by eye and see if there are any unusual patterns.
    For example, maybe all 4 aces are in the middle 13 cards in half of the
    winning sequences.
    
    I would be surprised if there was nothing unusual about the patterns
    of winning decks.  But what those patterns are, I don't know.

      John