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Figuring Out Sequential Royals

Ten years ago, I used Video Poker Tutor as a computer trainer. Among other things, it allowed you to calculate the value of sequential royals. It would also take 20 minutes to figure out how muchBob DancerBob Dancer is one of the world's foremost video poker experts.  He is a regular columnist for Casino Player, Strictly Slots, and the Las Vegas Review-Journa land has written an autobiography and a novel about gambling.   He provides advice for tens of thousands of casino enthusiasts looking to play video poker.  Bob's website is  a game was worth.
Today, WinPoker does not allow you to consider sequential royals, but it takes 20 seconds to figure out how much a game is worth. A lot of the increase in speed is the increase in computer processors during the past ten years, but a big chunk of the increase comes from short cuts. And the elimination of sequentials was part of the price for that increased speed.
But you're planning on playing, say, 9/6 Jacks with a 50,000-coin two-way sequential, and you want to know what strategy variations you should consider, and you don't have access to the old Video Poker Tutor (which doesn't run on modern computers). How do you do it?
If you have 4-cards in sequential order, you have a 1-in-47 chance of completing a 50,000-coin hand. That is worth over 1,000 coins, so you should break a 250-coin (9KQJT) straight flush, so long as the royal cards you have are sequential.
For 3-card royals, in sequential order, you usually have a 1-in-1,081 chance of getting a royal. This makes the royal draw worth $3.70 to the dollar player, without including the sequential. But half of these are sequentials, you need to add $46,000/2,162 to this total, which adds another $21.28 to the value of a 3-card sequential royal. We now check WinPoker on a hand such as (QJT9)8 or QQ(QJT), and we discover that if we increase the value of the 3-card royal by $21.28 that it would be the best play. This means 3-card sequential royals are superior to dealt straights, 3-of-a-kinds, and 4-card straight flushes, but not dealt flushes.
For two card royals, you have a 1-in-16,215 chance of completing a royal. But only one in six of those are sequential. For example, starting from (AK), the possible royals including position are (AKQJT), (AKQTJ), (AKJQT), (AKJTQ), (AKTQJ), and (AKTJQ). Only the first of these six counts as sequential. So having two royal cards in sequential order adds $46,000 / 16,215/6 = 47.3¢. We now have to search each of the possible types of 2-card royals to see if adding 47.3¢ changes the play.
I took maybe five minutes to create the following list:
From (AK), (AQ), (AJ), I found them superior to all 3-card straight flushes except (QJ9) and (JT9), and all 4-card inside straights.
From (AT), I found it always superior to a ace by itself or two unsuited high cards. I also found it superior to AKJT, AKJT, and AQJT, so long as the fifth card wasn't a flush penalty. It was also worth more than 3-card straight flushes containing a gap.
From (KQ), and (KJ), since it is more valuable than (AK), (AQ), and (AJ), and you can't have (QJ9) or (JT9) in the same five cards without creating a higher-valued combination, then we have these combinations worth more than all 3-card straight flushes, including (KQ9) and (KJ9), as well as all inside straights.
From (KT), it is superior to (KT9), AKQT or AKJT (although it's a very close play when the fifth card is a flush penalty to the (KT). It is superior to all 3-card straight flushes with a gap, including A-low and (Q98).
I didn't work out (QJ), (QT), or (JT), but you get the idea.
For a 1-card royal, there's normally a 1-in-178,365 chance for any royal, and only 1-in-24 of them are sequential. That gives a 1-card sequential royal an extra penny in value. ($46,000 / 178,365 / 24). A "queen in the middle" receives twice as much of a boost because it may be part of both a (AKQJT) and an (TKQKA) royal.
To find these changes, I set WinPoker to "Hard Hands" with a value of 1¢ and turned on "autohold". I played it 150 or so times until I became convinced that I'd seen all of the hands where the difference between the two best plays is 1¢ or less. The only one that fit into this category is you prefer a sequential J to an unsuited AJ.
More important than the actual strategy variations, though, is the methodology of how to figure it out. I've tried to show that here, and it can be applied to any game, with a sequential of any amount, and whether the sequential is one way or two ways.
Many players, of course, don't want to figure it out for themselves. They'd be happy to spend $5 or $10 for an accurate strategy (and then share it with their friends for free), but it takes me far too long to do that for that price to be of interest. Also, sequentials are relatively rare these days.
But they do exist. And now you have the tools to analyze them.
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