# Mulligan!

We’re taking the mulligan. I have only had one response from Bart Wright and he doesn’t like starting with three Kings. Four guaranteed turnovers ain’t bad, but it’s only slightly above average, but you don’t need a math Ph. D. to work out three kings is way more than we deserve. I would agree with this analysis.

Normally I won’t take Mulligans since I like the challenge of playing all starting hands, not just the good ones. But I was willing to make an exception to test the reader’s skill of evaluating a start position. Bart takes the Mulligan and it would be fun to see to concept of “poor decisions and consequences” play out in practice.

Bart says he is only interested in winning, not score or number of moves. But it’s hard to get a strong consensus if Bart is the only player to comment. Hence I consulted my random number generator app on my iPhone. I will choose a random number between 0 and 1999. Any number 1000 or greater means we play to win with a score of 1000 or better.

Yep, looks like we’re playing for a score of 1000+.

We take the Mulligan. Our mission, which we must choose to accept, is to play this hand and clear all eight suits with a score of 1000 or better. That’s the bad news.

The good news is we get six turnovers, no kings, no aces. What’s not to like? Good call Bart!

The obvious choices are to shift the Two of Clubs or the Jack of Diamonds since these are in-suit builds and the columns contain 4 face-down cards. Normally building Q-J is attractive with a spare Queen, but here we also have a spare Jack in Column 9. So it’s a toss-up between shifting the 2c or Jd first. My RNG says shift the Jd.

Our good luck continues. We get no less than seven in-suit builds and we are still guaranteed at least two more turnovers. We haven’t seen an Ace or King yet, but we would probably welcome either card now. Experienced players know that one advantage of drawing an A or K in the first round is less chance of drawing four-of-a-kind Aces or Kings in the last deal!

Clearly our next move is to shift the big club stack onto either Nine of Spades. Remember that we are playing for a score of 1000+ so it might make a difference if we shift it to column 2 or column 8. Since we want empty columns asap it makes sense to move it to column 8, all other things being equal.

Before we proceed, here is a simple exercise for the reader:

• What is the probability that we can clear column 5 in four moves (even if it’s not optimal play)? Obviously, we need three good cards in a row. Give your answer to 2 decimal places and assume all ranks from Ace to King are equally likely (even though we have e.g. three Deuces and no Kings exposed)
• (ADVANCED) Remove the assumption of all ranks from A to K being equally likely. Write a computer program to simulate 10,000 iterations of this exact hand to estimate the chances of clearing column 5 within four moves.

## 2 thoughts on “Mulligan!”

1. Bart Wright says:

“Bart takes the Mulligan and it would be fun to see to concept of “poor decisions and consequences” play out in practice.”

Caveat: don’t try this in Las Vegas. We know there’s an awful lot of luck involved. Math will serve us better until we can do the same test at least 20-odd times, unless we get a really dramatic estimate like 1% or 99%.

“Bart says he is only interested in winning, not score or number of moves.”

I certainly didn’t mean to imply you ought to do things differently! I’m in Rome, after all. I’m just saying it’s going to be unfamiliar to my way of thinking and I’ll probably forget or have bad intuitions about that. For instance, you debate which 9 to move your 8 to — and to my ordinary way of thinking it makes absolutely no difference because you can move them later, with no effect on your chances of winning.

“Normally building Q-J is attractive with a spare Queen, but here we also have a spare Jack in Column 9. So it’s a toss-up between shifting the 2c or Jd first. My RNG says shift the Jd.”

They’re both excellent moves, but I think the J on Q is a tiny bit better and here’s why: Once you’ve made that move, and the next card comes up, all of the options for playing that rank are still there. You can still move a jack. If you move the 2 onto the 3, then a 2 coming up can no longer be played anywhere.

“What is the probability that we can clear column 5 in four moves (even if it’s not optimal play)? Obviously, we need three good cards in a row. Give your answer to 2 decimal places and assume all ranks from Ace to King are equally likely”

If we just paid attention to ranks, 5 let you keep going and 8 do not, or 0.3846153846. But you have to get that 3 times in a row, so the cube of that is 0.0568957669. But that’s too pessimistic, because some of the cards you can turn over give new options for turning cards over. I got 2 of the 5 providing new ranks you can move — but if the card you turned over was a jack, you can no longer move a jack. At this point it’s more complicated than I’m willing to do.

What I did do was look at what happens when we DO consider how many cards are unplayed in the various ranks. There are 87 cards we haven’t see yet, and they can be broken down by rank. The way it looks to me is 36 of 87 cards can be moved, or 0.41379310344 success for one round. If you need to do THAT three times in a row, it’s 0.070851613, but once again that’s too low.

I’m not double-checking all these things because this is supposed to be fun. But I make even more mistakes than I used to.

(Note: all those decimal places coming off my calculator are meant to be a wry joke.)

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2. I’m keeping up so far. But I suspect things will get more complicated soon.

Bart, an excellent joke.

The main probability I am concerned with now is what’s the chances of me foregoing your math exercise and heading to the veranda with a cold Pacifico? I will give earthquake and heart attack a generous portion of .0003%, leaving me with a 99.9997% chance of success.

I will try to do a coupla’ posts a day and maybe catch up with the top of the page soon.

As an aside note, I am recovering from a ‘puter krash-n-reload-start-from-scratch-&\$@^**\$@# and what should pop up but Microsoft Solitaire Collection !! So I am running out of excuses.

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