In the last week I asked the following question: how many rows do we need to deal from the stock to be sure of procuring an empty column (assuming the worst possible permutation of unseen cards)?
First let us clear up the Captain-Obvious stuff: Column 2 is the only column with no unknown cards so we must focus on that. Also, there is not a single Nine anywhere until the second deal so the answer must be at least 2. By that time, two Sixes will appear in Column 2 so we need to find enough Sevens to take care of these Sixes.
Schistocerca Americana gave a correct answer of three rows. I say correct because the Grand Faster mucked up by not asking for the minimum number of deals. I should have asked what is the minimum number of rows we need to deal from the stock to be sure of procuring an empty column?
Bart gave another correct answer of two rows. Starting from the game state from last time:
The following moves do the trick:
Before deal: dj,aj
After deal 1 (J6254A388K) : db,ad,ba,ge
After deal 2 (562259AJ8Q): gc,bg,bf
Note that the first move dj is a typical tesuji (link) when playing with undo. This can only be explained by prior knowledge of cards in the stock – it is inconceivable an expert player can find some miniscule advantage of dj over “doing nothing” if playing without undo. Also observe that we got lucky with ba after deal 1: the Five and Six are the same suit, hence the move is indeed legal.
Of course, it will be desirable to achieve an empty column without dealing any rows from the stock. We can guarantee at least three turnovers in columns 1,6 and 7. On a good day, we will get an empty column without any of the shenanigans described above. The worst-case scenario says we are forced to deal two rows, take the empty column and proceed from there.
Our luck is in: the final hidden card is the Queen of Spades which can immediately go onto the King of Hearts in column 5. So now we know it is possible to get an empty column without dealing any of the shenanigans described above.
Our cheat sheet now looks like the following:
The power of an empty column should be pretty clear. For most of the columns it is easy to determine the next face-down card, then undo to recover the empty column.
It is time for a new question: Is it possible to determine the identity of the next unknown card for all columns containing at least one face-down card?
Assume we are allowed to restart from the very beginning, but cannot deal any cards from the stock.
Thanks to Bart Wright and Schistocerca Americana for once again reminding me of my lack of cultural knowledge (e.g. Kung Fu). There is only so much one can do with my favourite animal types from Phil Hellmuth’s book Play Poker Like The Pros 😊