Dealing with Early Kings

Intro

Below is an actual game example.

Over the many thousands of games that I've played over the years, this has to be one of the most bizarre. In both good and bad ways, it defied the odds.

As I always do, I played this game without undoing moves or replaying the game, and it was not a previously-won game.
Kings are very often troublesome, whether early, late, or anywhere between. In this case, two Kings came out in the pre-game deal.

Two Early Kings

The first round went fairly well. I'd managed to turn 13 hidden cards, a bit better than my average game and a bit fewer than my average win. But, since only pile 2 contained suit breaks, I'd removed all the hidden cards in two columns, and hearts were mostly built. It was a very decent start.

Since I know all too well have many problems Kings can cause if not dealt with, I decided to forgo moving the Seven out of pile 6 and remove one of the Kings instead. (At this point in the game, with so few cards yet in play, the chances were good that the card underneath the Seven could not be placed. No matter how I played it, there was a good chance that I couldn't vacate a column after the deal.)

It's a technique that I've used many times. It often works. I had not a lot to lose since, in total, the Kings were blocking 9 cards, which would almost certainly become a major problem to deal with later in the game. Since the hearts looked like they might eventually be built in pile 9, and because there were 5 cards in pile 4 under the King, I moved that King of clubs into column 8 and initiated the first 10-card deal (not counting the pre-game deal before the start of the game).

I should add that there was another very important reason for moving the King of clubs instead of the King of hearts. Once a King is placed into a vacate column, it's there to stay until either its suit is built or the end of the game. This makes the King the perfect head for a garbage pile. It would be expected then that, eventually, the pile would contain several suit breaks and possibly some breaks in rank as well. Since the King of clubs was alone, it could later be built up. However, the King of hearts ran down to the Nine, leaving four cards less room to build upon.

King Extracted

As is typical and expected, I failed to vacate a column after the deal. Still, I managed to turn several hidden cards. Following the second deal, I emptied a column, turned several more cards, and even managed to move the King of hearts from pile 9 into column 10, leaving a King-Two suited run.

Then, a very strange and highly unlikely event occurred. On the third deal, the Ace of hearts landed squarely on pile 10, which completed the build and vacated the column. The only other time this has happened to me that I can recall was years ago, and it did not vacate the column. I continued turning cards, even built the suit of spades. Things were going well. It looked like an easy win from there on in.

The fourth deal was not so good. I turned a few cards and managed a bit of ordering, but could do no more. By this point, things still weren't bad. There were only 9 hidden cards, lower than for my average win.

Just prior to the final deal, there were four or five good chances at vacating a column after the deal. Based on my past experience, I would estimate that my odds of winning at that point were greater than 95%. I thoroughly expected to win, especially since I very rarely lose after completing two builds.

The Final Deal

Then, another very strange and highly unlikely event occurred. After the fifth and final deal, there was no possible way to vacate a column or turn a hidden card. Why? There were a few reasons.

The biggest reason was no doubt because the deal was disgustingly terrible. It contained two Kings and two Aces, as well as two useless Nines. Of the 9 hidden cards remaining, 4 or them were Fours. Not a single Four came out in the deal, and there were three, maybe four, Threes waiting to be loaded. It was not so much that I readied the Threes but that there was just no place to put them.

But, as bad as the deal was, if the Nines in piles 9 and 10 that had different suits have landed reversed, I could have vacated a column and most certainly won the game. The cards were there. I just couldn't get to them.

After undoing some moves, I was able to beat the game, but it wasn't easy. It's very rare to get such a horrible final deal when things are fairly well ordered. The victory was not as satisfying as it would have been without undoing, but I'll take what I can get.

Lessons Learned

Now, what are the lessons here?

1. Whenever possible, if a King is blocking access to more than a few card cards, whether they are hidden or not, consider removing the King. Ideally, this should happen when more than one column is vacant, but in the middle and late rounds, one empty column often works, and under favorable circumstances, even before the first initiated deal.
2. Normally, it's not good to remove a King before the first deal unless more than one King is blocking several cards. Even then, the player should be very careful. It can and often does backfire. But, since the changes are great of a loss anyway, the act of desperation might pay off.
3. When a King is moved into the only vacant column before the first initialized deal, the player should anticipate having a difficult time vacating a column after the second deal, and perhaps even the third.
4. A game that is going well should never be considered won until it mathematically can't be lost. Terrible deals can happen and often do. The player should never take a game for granted and should be careful not to miss good moves. (In this case, however, it was pure bad luck, not sloppy play.)
5. On the other hand, favorable deals can and do happen as well. One good round can make up for one, two, or even three bad previous rounds.

Addendum

Incidentally, I've computed that with two decks of standard playing cards, the chances of any particular rank (say, the Four) having a count of 4 or higher in 9 randomly chosen cards is very close to 1 in 615. The chances of any rank in general, is about 1 in 47. (Since I've computed these values both mathematically and via simulation and got very similar results, I tend to believe that the quotes are accurate to the closest integer.)

So, we should anticipate that in the long run, in about 1 out of every 47 games, when only 9 hidden cards remain, that 4 or more of them are the same rank. Yes, I've seen it happen several times. I've seen it happen with Kings. I've beaten it with Kings without undoing a single move. It can be done! You can do it too.
The thing is, usually, these types of games will be lost and the player will never know why.

Last tip of the day: If a player doesn't know why they lost, it's always a good idea to attempt to decipher why?
Then, if they can, they might modify their play to reduce the chances of losing as often in future similar situations.

One quick method of calculating the probability quoted above is shown below in Python3. To get the probability for any rank in general, simply divide the output value by 13.

from math import factorial as fact
nCr = lambda n,r: fact(n)//fact(r)//fact(n-r) #n choose r
m,n,r = 4,104,9; k=n//13
1. / (sum([nCr(k,i)*nCr(n-k,r-i) for i in range(m,k+1)]) / nCr(n,r))
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Output: 614.8187641608681