```Hi Eric,

That's interesting. The understanding I had was that a dealer could
're-value' their hand with each card dealt; that is - if they draw two
aces they're obviously 11 and 1 and they have to hit again. If they
then draw a card which would cause them to bust, they can treat both
aces as 1.

I don't know this for definite, except that the description I've seen
for what a dealer does did not specify what order the ace appears in.
Maybe someone knows what really happens....

/dh

On 6 Jan 2008, at 20:34, Eric I. wrote:

> On Jan 6, 12:02 am, "Eric I." <rubytrain... / gmail.com> wrote:
>> Although very close, my results do differ slightly from yours.  For
>> example you determine that when an ace is the upcard, there's a
>> 36.07%
>> chance of getting 21 exactly.  I get 36.35% (31.07% natural + 5.28%
>> "unnatural").  On the other hand we both get a 21.32% chance of
>> busting when the upcard is a 10 or facecard.  It'll be interesting to
>> see why that's the case when we post our code solutions.
>
> It took me a while, but after comparing Denis' code and my own, I
> figured out where the differences in results came from, and it was in
> how aces are handled when valuing a hand.
>
> I believe that Denis uses an incorrect algorithm.  When valuing a
> hand, he first sums up the values of all the non-aces and then makes a
> second pass handling the aces.  For each ace, if valuing it as 11
> would *not* bust the hand, he values it at 11.  Otherwise he values it
> at 1.
>
> But consider a three-card hand like ace, ace, 10.  The first ace is
> counted as 11 since that wouldn't bust the hand.  The second ace is
> counted as 1, since valuing it as 11 also would bust the hand.  But
> the hand still busts due to the 10.  If both aces were counted as 1,
> then the hand would not be a bust (so far, at least), and it would
> need for another hit.
>
> When I changed Denis' hand valuation logic on aces to my own, our
> results were identical.
>
> Eric
>

```