I was watching a Nick Sibicky lecture (I forgot which), and he mentioned odd-numbered extensions are more advantageous than even-numbered ones. I'm all for implementing this in my gameplay, but I'd also prefer to know the reason for this.

Never heard of “odd-numbered extensions” (not that that would mean anything ), but maybe you mean this?

From http://senseis.xmp.net/?ExtensionFromAWall

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“The only difference between me and a madman is that I’m not mad.” — Salvador Dali ★ Play a slooooow correspondence game with me on OGS?

The reasoning is the potential for creating eyes ...

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The really most difficult Go problem ever: https://igohatsuyoron120.de/index.htm
Igo Hatsuyōron #120 (really solved by KataGo)

Sounds suspicious.

And then there is Lockett's Law: http://senseis.xmp.net/?LockettsLaw

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The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Side extensions from single corner stones are covered somewhat in this set of workshop notes by Yilun Yang. In general, there is no hard-and-fast rule, as it depends on the surrounding stones, whether you seek to expand or reduce, and other tactical and strategic considerations.

The rule doesn't make sense. For example, there are several cases where a 2-space extension is great. Maybe 4-space extensions are less common, but they can happen, too.

Anyway, there is an "ideal" extension for any given situation, but this is hardly quantifiable into a rule like this.

I would say that in actual play, 2, 3, and 5 space extensions seem more common to me than 4 space extensions, but this isn't even true for mini-Chinese opening formation, where there's a 4-space extension made quite early by black.

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be immersed

I discovered a new law: "You should always extend a prime number of spaces". I'm sure 7 space extensions are great too, not to mention 11-space extensions

11-space extensions are usually referred to as nirensei.

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Still officially AGA 5d but I play so irregularly these days that I am probably only 3d or 4d over the board (but hopefully still 5d in terms of knowledge, theory and the ability to contribute).

I started to say the same thing as Kirby about 4 space extensions. I feel like there's something awkward to them.

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Occupy Babel!

Pecentage of games including a 4-point extension by Dosaku as white on the 3rd line: 45/124 = 36%.

I guess confusing worked for him.

Extensions from what? Otherwise this thread (and the original statement) are nonsense.

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Dave Sigaty
"Short-lived are both the praiser and the praised, and rememberer and the remembered..."
- Marcus Aurelius; Meditations, VIII 21

OP is smart to ask why. He is following another famous proverb, "Beware following proverbs you don't fully understand."

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- Brady
Want to see videos of low-dan mistakes and what to learn from them? Brady's Blunders

When extending, as with any move, ask yourself what you want to accomplish

- making a firm base, which cannot be disconnected but which might become overconcentrated?
- making a loose base, which can be disconnected but which allow you to move faster while sacrificing some stones?

This all depends on the circumstances, as the extension (from a wall) will also have a function towards the other side

It is better to think of moves this way then to follow rules like the one mentioned. They are restricting your thinking at best and misguided at worst.

If I only made moves I fully understood, I would not play at all.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.