Archive for December, 2009

Searching a Hex

December 3, 2009

(Inspired by Chgowiz’s “Finding things in a 5 mile hex”)

So there’s basically four variables in the equation.

1. How big the hex is
2. How far they can look
3. How fast they can move
4. How likely it is that they recognize the target when they see it.

1. How big the hex is:

Unlike labyrinth maps, wildness maps are usually recorded on graph paper with hex grids, at a scale of 6 or 10 miles for each hex. -LL pg 45

Assuming this is ‘centre to centre’ distance, which is the ‘short width’.

“t” is the length of one of the hex’s six edges:

>> Hex Width (opposite sides, aka short diameter) = 1.732t

For our example, we divide 6 by 1.732 and get 3.464 miles.

>> Hex Area = 2.598t^2

So the area of our example hex is 31.178 square miles.

This is more useful in square yards (3,097,600 square yards to one square mile):

96,576,443 square yards. Yes, almost 100 million square yards.

2. How far they can look:

when monsters are encountered the Labyrinth Lord will roll 4d6 x 10 to determine how many yards away the characters are from the monster. -LL pg 50

Assuming weather conditions are normal, other ships can be seen when up to 300 yards away. -LL pg 58

4d6 averages to 140 yards.

The party can decide to use whatever “search footprint” they like, though. A large value is a quicker but less thorough search, while a small value is a slow but methodical search.

Whatever value they choose (called the Search Radius), the search footprint is 2*pi*r:

For 140 yards, this is 61,575 square yards.

Note this is per observer: If four characters spread out 140 yards apart along the search line, then their search footprint is four times larger (and help is that much farther away if a wandering monster attacks.) Furthermore, there are fewer characters to spot a hidden or secret target.

It’s useful to figure out how many search footprints are contained within the hex, simply by dividing the area of the hex by the area of the footprint:

For our example, 96,576,443 / 61,575 = 1,568.4 footprints. Expressed as a percentage, the odds of a terrain feature falling into any given footprint is about 0.064%.

3. How fast they can move:

A character that moves at 120 (feet or yards, depending on environment) can move 24 miles in the wilderness per day. -LL pg 45

Given our search footprint, how many footprints can a character move through?

This depends on the character’s inherent movement rate and also the Terrain Movement reduction amount, if any.

Determine the modified movement rate, and then divide by 12 to get a movement rate per hour assuming 12 hours of marching and 12 hours of rest.

An unencumbered human moving through the plains moves at 2 miles per hour, or 3520 yards per hour.

Divide that by the diameter (twice the radius) of the search footprint to determine his footprints/hour rate.

For our example: 3520/280 = 12.57 footprints/hour.

This can be multiplied by the number of independent searchers, so for a party of four we have about 50.3 footprints/hour.

Hours are not necessarily the most useful unit of time here.

Wilderness Wandering Monsters: This check is only made 3 to 4 times per day of game time in wilderness adventuring. –LL pg 125

4 times per day is once every six hours. Our party is searching for 12 hours a day, so six hours is a useful division of time (perhaps as “morning” and “afternoon”.)

Multiplying by six hours gives us 75.4 footprints/quarter-day per person, or 150.85 footprints per day of searching.

That is close to 10% of the hex’s footprints per day, or 5% per quarter-day.

Thus, for our example, based on these rates of searching, every searcher has a 5% chance every six hours to run across the terrain feature. A generous DM might make this cumulative.

4. How likely it is that they recognize the target when they see it.

The first thing to test is whether the party chose a sufficiently tight search pattern.

The potential target-finder should roll 4d6 and multiply by 10 for a monster-sized target or by 20 for a ship-sized target.

If the roll is less than the search radius, the target is not found after all — it has slipped through the coarseness of their search pattern. Otherwise, the target has been found, but it might not be recognized.

If the target is not hidden in any way, then the character has found it. Otherwise it is detected only on a 1 in 6 chance, as a secret door. Elves and dwarves might find it on a 2 in 6 chance, if it is related to their racial abilities. Thieves may use their Find Traps skill percentage.

A failure here means the searchers have walked right past their target; they must complete their search of the rest of the hex before trying again.