Does a skill in Burning Wheel open gray if one of its root stats is gray?

I'm using the BW Gold rulebook. I'm trying to start a supernatural character, and here's what it has to say about skills and their root stats:

New skill exponents start at half the root stat rounded down. If a skill has two roots, half the average of the root stats rounded down. (p87)

Great, this covers skills of the same shade. Later, it says:

It costs five skill points to open a skill at its root with a gray shade. (p88)

However, this evolves into a contradiction. Later in the rulebook, it says:

When a stat acts as a root for a skill, the skill takes on the shade of the stat. If the root comes from the combination of two or more stats, the shade is the darker of the two. (p545)

Except the given example somewhat contradicts the previous rule:

A character with a G5 ability and a B5 perception opens a Surgery skill [Perc/Agil]....Normally, half of the average of 5 is 2.5... [rounding] to a root of 2. However, with a gray stat, the math is different: Agility counts as two greater because of its shade. So the actual numbers to average are 7 and 5. (p545)

This seems to directly contradict the previous example. So which is it? There are three cases for two root stats:

• Both stats are black: This obviously just averages and rounds down; shade black.
• One stat is black, another is grey: I'm not sure what happens here. The above examples contradict - should I +2 the grey stat for calculation, or just adopt the higher shade?
• Both stats are grey: From the Heroic and Supernatural Stats section, I would probably just set the skill grey. However, the Character Burner disagrees, indicating that I still have to spend 5 points to get a grey skill. What should I do here?

and two cases for one root stat:

• The stat is black: This is pretty obvious. Cut in half and round down; shade black.
• The stat is anything else: This runs into the same contradiction as above. The Character Burner claims it takes 5 points to make the skill grey, but later in the book, this appears to be a contradiction. What should I do here?

So, I'm confused. How should I handle these cases? Normally I wouldn't quibble over rules like this, but these rules are pretty critical, vastly changing probabilities and balance.

The first two quotes are from the Character Burner, so may not be intended to deal with grey stat/shades. Still, it's pretty confusing.

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Those lines don't contradict each other. It opens at the darker shade, and the total before averaging is boosted by two for having a gray-shade root involved.

So a G5 agility and B5 perception will open Surgery…

• in the darker shade: i.e., black…
• at half the average of the roots, with a lighter shade counting numerically as +2: i.e., 3…

… therefore, G5 Ag and B5 Per gives a B3 Surgery, which is the same as B7 Agility and B5 Perception giving B3 Surgery.

Contrast this with B5 Ag and B5 Per, giving a B2 Surgery; or G5 Ag and G5 Per giving a G2 Surgery. Notice how B3 Surgery is better than B2 Surgery but worse than G2 Surgery.

Note that the averaging of shades and the averaging of numbers are independent, although they key off of shared data points. So you average the exponents then halve them, and the averaging equation gets a boost if the shade mismatches. Then you average the shades themselves. Then you take those two things and it gives you the shade and exponent of the skill.

But how does "averaging the shades themselves" work?

According to "Gray and White Math", Monster Burner, p. 367:

• To mix white with darker shades, add 3 to the total before averaging (instead of 2 for mixing gray with darker shades). (It is implied that this is a one-time boost for the entire averaging equation, not added for each white or gray shade, probably to keep it from getting ridiculous.)
• To find the shade when multiple shades are involve, literally average them and round darker. So white plus gray is gray; white plus black is gray; gray, gray, and black is black; white and black and black is black.

The third moving part of this is that you can pay 5 extra skill points to open a skill at gray shade instead of black. This is entirely independent of gray-shade math, so it isn't a contradiction, it's just an alternative. You could get a gray skill by opening it off a gray stat, but you don't have to: you can open a skill off a black stat, and pay 6 points to get a gray shade (1 to open, 5 for gray shade).

For example, with a B4 Agility you would normally open a B2 Sword for 1 point, but by spending 6 points you would open it as G2 Sword.

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Monster Burner, page 368

White and white make white. White and gray make gray. White and black make gray. Gray and gray makes gray. Gray and black makes black. Black, white and gray make black. White, gray and gray make gray. White, white and gray make gray.

So, a skill with black and gray roots makes a black skill.

However, when mixing gray with black, add 2 to the equation before averaging. Then half rounded down, as per normal.

G5 Ability and B5 Perception = [(5 + 2) + 5] / 2 / 2 = 12/4 = B3 Surgery

White is similar, but you add 3 when mixing with gray or black.

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