Ask The Wizard #419

First, let me clarify some video poker terminology.

• Sequential Royal = Low to high only (10-J-Q-K-A)
• Reversible Royal = Both directions (10-J-Q-K-A or A-K-Q-J-10)

To make matters more confusing, not everybody uses this terminology and a game that has a separate line item for a sequential royal might pay both ways. Hopefully the rule screens would clarify that.

That said, the easy and perfect way to play a hand that has a possible sequential or reversible royal is to put it into my video poker calculator. Here are the links:

• Sequential Royal calculator
• Reversible Royal calculator

If you can't or don't want to use my site at the machine, note here are the average wins for a royal when a reversible royal pays 50,000 for a 5-credit bet according to how many cards are already in position.

• 4 to a reversible royal = 10,000
• 3 to a reversible royal = 5,400
• 2 to a reversible royal = 2,333
• 1 to a reversible royal (middle position) = 1,567
• 1 to a reversible royal (not middle position) = 1,183
• 0 to a reversible royal = 800

For sequential royal games (that pay low to high only) use the 1,183 figure regardless of position.

With these figures, use my video poker strategy maker and create a separate strategy for each one of these average royal wins. Then use the strategy at the machine according to how many cards to the royal are in the correct position.

This question is asked and discussed in my forum at Wizard of Vegas.

No!!!!!!!! No, they did not do the math correctly. In fact, the sphere is the WORST three-dimensional shape if they want to maximize the ratio of surface area to volume.

Let's start by looking at the surface area equations on the back of the box.

They do correctly state the surface area of a sphere, or what they would probably call a donut hole, is 4πr², where r=radius.

However, they incorrectly state the surface area of a torus, what they would probably call a donut, is 2π²rR. The actual formula is double that or 4π²rR Please refer to the following image for r and R.

r = radius of red circle
R = distance from the closest part of the torus to the center.

Image source: Wikipedia page on the torus.

You might say the alternative to a sphere is only applying glaze to half the torus, like frosting on a donut. However, I submit a careful look at a torus-shaped traditional Apple Jacks shows they apply glaze to the whole thing.

To make my next point, let me also provide the formulas for the volume of both a sphere and torus.

• Sphere = (4/3)πr³
• Torus = 2π²r²R

As a reminder, the formula for the surface area is the derivative of the volume.

For a torus of r=1 and R=1, we get a surface area of 39.478418 and a volume of 19.739209. The ratio of surface area to volume, or the glaze ratio, is interestingly exactly 2.

To equate the volume, the radius of a sphere would have to be 1.676539. For that radius of a sphere, we get a surface area of 35.321350 and a volume of 19.739209. The ratio of surface area to volume, or the glaze ratio, is 1.789400.

In other words, the torus delivers more surface area or glaze for the same volume.

I mentioned earlier the sphere is the worst three-dimensional shape to choose of the goal is to maximize the ratio of surface area to volume. This is known as the Isoperimetric inequality. While this has been proven, I think it's self evident. For example, bubbles strive to minimize surface area and maximize strength and they are spherical in shape.

The bottom line is you should get the spherical or donut hole shaped version of Apple Jacks or anything else if you want to minimize glaze, not maximize it. That would be my goal as I find these cereals much too sweet and would welcome less glaze. I also absolutely call out Kellogg's for false advertising for which they get the rare Wizard finger-wagging of shame.

This question is asked and discussed on my Wizard of Vegas forum.

For more information, I recommend the video Internet spots big mistake on Kellogg's cereal box YouTube video by Presh Talwalkar of the MindYourDecisions channel (one of my favorites!).

I can see how the answer to this question would have practical application in some video poker variants that offer a bonus feature if the player gets a winning hand on the deal.

The answer depends on the format of video poker. The following table shows the number of combinations and probability for all possible events on the deal in video poker with a 52-card deck with no wild cards, starting with a lowest paying hand of a pair of jacks.

The probability of any winning hand in jacks or better video poker games is 0.206275.

The second table shows the number of combinations and probability for all possible events on the deal in video poker with a 52-card deck where deuces are wild, starting with a lowest paying hand of a three of a kind.

The probability of any winning hand in deuces wild video poker games is 0.184394.

The third table shows the number of combinations and probability for all possible events on the deal in video poker with a 53-card deck, including a joker, starting with a lowest paying hand of a pair of kings.

The probability of any winning hand in joker poker (kings or better) video poker games is 0.195845.