One and a half versus the square root of two

After looking at diagonal movement on a square grid, I started wondering how far a figure would need to move before the difference between ³⁄₂ and the √2 square root of two became significant. (Simple curiosity here. Nothing more.)

What amount of error is significant? ¹⁄₂? 1? ³⁄₂?

Let’s go with 1. Check my math: (sqrt(2) * x) + 1 = 1.5 * x

It turns out x is about 12. Here’s my check: (1.5 * x) - (sqrt(2) * x); x = 12 which yields a result just a little over 1.

That’s smaller than I was expecting. Though it is still reasonable for most games. A twelve square per turn movement rate will start to get problematic for other reasons.

More zero dice

So, this was written last year. (16 June 2010) as a follow-up to “My favorite die”. Seems I never got around to publishing it...

I have some six-sided dice marked zero to five.

I believe I got these out of the bin of dice at Kaleidoscope Toys.

The only game I know of that uses such dice is a “WWII space combat” game called Hard Vacuum. (Which I haven’t played.)

They create some nice ranges. Two yields 0–10. Three, 0–15. Four, 0–20. (Keeping in mind, of course, that summing multiple dice doesn’t give linear results.) Use one which a d10 (or d20+) to emulate a d60.

Dice with a zero are also nice for open-ended rolls, as it means you don’t have numbers that can’t be rolled.

And, if you don’t have any 0–5 dice, just roll a normal d6 and read 6 as zero.

Diagonal movement on a square grid

In games, how do you handle diagonal movement on a square grid?

One of the easiest methods is to simply not allow diagonal movement. In the image below, the circle in the lower left indicates a figure to be moved. Red shows which squares it could move to with one move. Orange, two; yellow, three; green, four; blue, five; & violet, six.

(I only drew the northeast quadrant, but the other four quadrants would look the same.)

Another easy option is to simply allow a diagonal movement to cost the same as orthogonal movement. Which looks like this:

Neither of those are very satisfactory, though.

The d20 system (WotC 3e D&D) has the cost of diagonal movement alternate between 1 and 2. Thus it averages out to 1.5, which is close enough to the real distance of a diagonal move. (The square root of 2.) It ends up looking like this:

This year at the North Texas RPG Con, I picked up another way to handle it from Jeff Dee. Diagonals cost the same as orthogonals but cannot be consecutive. You have to have at least one orthogonal move between two diagonal moves. At first, I thought this was equivalent to the d20 system rule, but it is a bit less accurate. As shown here:

I also recently picked up a rule for estimating the distance to a flying target, though I forget where I found it. Considering the horizontal and vertical distances, add half the shorter to the longer. e.g. A flying monkey is 40' away and 20' up. So, use (20 / 2) + 40 = 50' as the range.

AoO & Move & Run

In the 3.5 version of the d20 system, there is some possible confusion around the Run and Move actions and Attacks of Opportuny (AoO).

(For those readers who don’t like the d20 system or AoOs, I suggest you stop reading now.)

First, there is a general rule: Leaving a threatened square draws an AoO with two exceptions. The first exception is if your only movement that round is a 5 foot “step”. The second is if you are using the Withdraw action.

Second, there are tables that indicate whether an action itself draws an AoO. Here it is indicated that both the Run and Move actions provoke AoOs.

This can be interpreted to mean that entering a threatened square during a Run or Move action draws an AoO. Because, if the first rule covers the leaving case, there’d be no need for the second rule.

On the Wizards’ web site, however, can be found this clarification:

The move and run actions are worth a special note. According to Table 8-2, both these actions provoke attacks of opportunity; however, the basic rule for movement and attacks of opportunity still applies. When you move from one square to another in combat, you provoke an attack of opportunity when you leave a threatened square—not when you enter a threatened square.

(Thanks to Von for finding that.)

So, for me, that makes it clear that Run and Move are in no way exceptions to the basic rule. The “Yes” in their AoO column merely distinguishes them from Withdraw. Just stick to the basic rule. Mostly...

There is, however, at least one exception to the basic rule: Crawling (which is covered by the Move action). Crawling is only a 5 foot move, and it provokes an AoO for both leaving a threatened square and entering a threatened square.

Zones of control & attacks of opportunity

(If you aren’t a fan of the d20 system, you may want to bail now.)

The d20 system justifies attacks of opportunity (AoO) thusly:

Sometimes a combatant in a melee lets her guard down. In this case, combatants near her can take advantage of her lapse in defense to attack her for free. These free attacks are called attacks of opportunity.

Hogwash! Yes, this is a role-playing game, and that’s not a bad way to rationalize it “in character”. The truth, however, is that this is primarily a game mechanic. It’s better—in my opinion—to understand the real reason AoO exist and that the description above is simply a façade.

The AoO rules are really a form of zone of control (ZoC) rules that were used in war-games long before role-playing games. Many role-playing games have included such rules as well, but they are often simplified enough that they don’t stand out the way that they do in the d20 system.

Indeed, within classic D&D can be found the origins of the d20 system’s Withdraw action, five foot step rule, and five foot threat range. (Also note that, in classic D&D, missile weapons cannot be used while in melee.)

Now, back to the d20 system. Let’s say we’re playing a d20 game without ZoC or AoO rules. An orc stands in the middle of a 15 foot wide corridor. Ten feet away stands a fighter, and a mage stands behind the fighter. The orc wins initiative and can move past the fighter, past the mage, and attack the mage. Even worse, if you’re using the optional facing rules (from Unearthed Arcana), the orc would be attacking the mage from the rear.

(Click on images for a bigger version.)

If this were “real life” (replace the orc with a generic bad guy), the fighter and the mage aren’t going to let the orc walk past them to attack from the rear.

With typical ZoC rules, the orc would have to stop his movement for that round once he moved into melee range of the fighter. (In fact, the first d20 Star Wars rules worked this way.)

The d20 designers wanted to replace restrictions with consequences. (That is a common theme in the d20 system.) So, instead of a simple ZoC that stopped an opponent’s movement, they needed a consequence for violating a ZoC. The consequence they choose was the risk of suffering a free attack, an AoO.

AoO also proved useful to provide consequences instead of restrictions in other cases. Many games simply say that you can’t fire a ranged weapon when in melee range of an opponent. The d20 system, however, makes it draw an AoO.

Now, I’m not going to say that AoO was the best choice. I might have preferred that instead of AoOs, actions be restricted with skill checks being able to overcome the restrictions. Similar to how the Tumble skill allows characters to move through an opponent’s square. But then, the way I prefer to run combat, I generally don’t need ZoC rules. Anyway, my point is that the AoO rules should be understood as a kind of ZoC rather than a lapse in defense.

ZoC and AoO rules only go so far, however. Put the orc, fighter, and mage in an open area; and now the orc can avoid the fighter’s threatened squares to attack the mage.

This can be mitigated—to an extent—if the fighter previously readied a move triggered by the orc’s movement, but it’s—in my opinion—a needlessly complicated affair.