June 29, 2011 · File under Fun and games

Design your sign-based grinder floors as 1-wide strips with 2 spaces between them. Or possibly a grid of 2x2 blocks.

I've done some analysis of sign-based mob grinder design in Minecraft. The results surprised me, so I'll give them here.

For background on the kind of thing I'm talking about, watch this video, but the basic idea is to build surfaces for mobs to spawn on with signs placed on the edges, which tricks the mobs into walking off and falling to their deaths.

In that video, the spawning surfaces are laid out as 3xL strips (where L might be 14, at a guess), with 2-wide gaps between them. What I've done is try to compare the efficiencies of various alternative layouts. The layouts I'm considering are 1) a grid of NxN blocks with 2m between them, and 2) arbitrarily long N-wide rows of blocks with 2m between them. For both cases, N = 1, 2, 3 and 4 are considered.

The simplifying assumptions I've made for this analysis are:

- A 2m gap is sufficient to allow mobs to fall
- Mobs move in each of the cardinal directions with equal likelihood
- Mobs can spawn on a single block (e.g. spiders don't require a 2x2 surface)
- The surfaces are infinite; effects at the walls of the grinder are ignored
- Mob movement can be modeled as a series of independent steps in a random direction. (This is kind of a big assumption.)

Here are the calculated values of:

- d = density: spawnable surface per unit area, affecting spawn frequency
- T = average number of steps taken before a fall on a randomly chosen spawnable block
- F = T/d: inefficency (T, considering density) the layout with the lowest U should drop the most mobs)
- s = sign density: signs required per unit area
- U = F*s: sign-usage weighted inefficiency (lower is better), for builders who want to use the least signs. These things use a lot of signs.

```
```
Layout
d
T
F
s
U
1x1
11%
1
9
44%
4
1x∞
33%
2
6
67%
4
2x2
25%
2
8
50%
4
2x∞
50%
4
8
50%
4
3x3
36%
6
16.67
48%
8
3x∞
60%
6.67
11.11...
40%
4.44...
4x4
44%
4.833...
10.875
44%
4.785
4x∞
67%
10
15
33%
5

The 1x∞ layout drops the most mobs per unit space (has the lowest F value).

If the best efficiency per sign placement is paramount (placing the signs is truly tedious), all the 1x and 2x designs are tied.

Mobs tend to move in a single direction for some period of time, so the assumption that mobs move randomly from block to block probably gives an unfair disadvantage to the NxN designs. Thus, it's not unlikely that the 2x2 design is the most efficient in practice.

The Nx∞ layouts have the additional advantage that they are easier to build.

Perhaps peaceful mobs might require only a 1-wide gap to fall, which would allow for more efficient designs.

I'll give the calculation of the values for the 3x∞ case as an example.

Let A be the average steps before falling for the middle row, and B be the same for the edge rows. so the grinder layout looks like:

... ... ...BBBBBBBBBBBBBB... ...AAAAAAAAAAAAAA... ...BBBBBBBBBBBBBB... ... ... ... ... ...BBBBBBBBBBBBBB... ...AAAAAAAAAAAAAA... ...BBBBBBBBBBBBBB... ... ...

A step in each direction is equally likely, so falling will take 1 more step than a step from each neighbor. Thus:

A = 1 + 2/4 A + 2/4 B B = 1 + 1/4 A + 2/4 B + 1/4 * 0

Solving,

A = 8 B = 6

Spawning is on a B block 2/3 of the time, and an A block 1/3 of the time, so

T = (A + 2B) / 3 = 20/3

Also, the repeating pattern is: space, block, block, block, space, with two signs, so

d = 3/5 = 60% s = 2/5 = 40%

and

F = T/d = 6.66... U = F*s = 4.44...

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