Leech

From Cities

Uses

Weapon: Leech
Damage	Attacks	To Hit	Break Rate
2(n-1)	×1	+5?%	?%

After attacking something, Leeches either become a larger version with doubled damage (you automatically switch to the new one) or drop away ("the leech is sated").

Your damage multiplier doesn't apply to Leeches, so they may be useful for leveling Pheonices.

Where to get it

Get low-damage leeches automatically by moving through a Swamp or Fire Swamp. Moving through Primeval Swamp will get you larger ones. Pick them up in any kind of Swamp for 5AP.

Dropped by monsters that can be found in Swamps, such as Crocodiles, even if they are in some other place.

Warning: Maths ahead

or Are leeches worth using for their damage alone?

Consider starting with n AP to spend. How much damage will you do, on average? Call this value d(n).

Then:

d(0) = 0
d(1) = 1

Now, with 2 AP to spend, your first leech can either do 1 damage and sate immediately (50% chance), in which case you have 1 AP to spend, doing d(1) damage, or it can upgrade (the other 50% of the time), and do a further 2 damage. So:

d(2) = (1+d(1))/2 + (2+1)/2

With 3 AP to spend, your first leech can do 1 damage and sate immediately, leaving you with 2 AP to spend (50% chance), or it can upgrade once and then sate (25% chance), leaving 1 AP to spend, or it can survive the whole experience.

d(3) = (1+d(2))/2 + (2+1+d(1))/4 + (4+2+1)/4

And so on:

d(4) = (1+d(3))/2 + (2+1+d(2))/4 + (4+2+1+d(1))/8 + (8+4+2+1)/8

And, in general (for n < 12):

        n-1
d(n) = sum ( 2i - 1 + d(n-i) ) / 2i    + ( 2n-1 ) / 2n-1
        i=1
                          n-1
     = n + 1 - 1 / 2n-1 + sum (d(n-i) - 1) / 2i
                          i=1
            n-1
     = n + sum d(n-i) / 2i
            i=1

Of course, after 12 successful attacks, your leech gets sated anyway, so the formula actually cuts off early. So, for n >= 12:

         11
d(n) = sum ( 2i - 1 + d(n-i) ) / 2i   + ( 212 - 1 + d(n-12) ) / 211
        i=1
                             11
     = 12 + d(n-12)/2048 + sum d(n-i)/2i
                            i=1

These formulae can be tabulated:

AP spend	Damage	Average damage / AP
1	1	1
2	2.5	1.25
3	4.5	1.5
4	7	1.75
5	10	2
6	13.5	2.25
7	17.5	2.5
8	22	2.75
9	27	3
10	32.5	3.25
11	38.5	3.5
12	45	3.75
13	50.9997558594	3.92305814303
14	57.0003662109	4.07145472935
15	63.0008544922	4.20005696615
16	69.0014648438	4.31259155273
17	75.0021972656	4.4118939568
18	81.0030517578	4.5001695421
19	87.0040283203	4.57915938528
20	93.0051269531	4.65025634766

Extending this computationally to 100000, the damage/AP value only just exceeds 6. The upper bound is (provably) 8192/1365, or approximately 6.0014652014652015. Thus, we can conclude that, unless you have a damage multipiler of 1 and no aligned weapons, leeches really suck. Darksatanic 18:49, 23 October 2007 (BST)

To get the upper bound, assume that d(n) tends to a limit of k. This means that, for sufficiently large n,

d(n) = kn

Put this into the generic formula above, and you get:

                         11
kn = 12 + k(n-12)/211 + sum k(n-i)/2i
                        i=1

Solve for k, and the n drops out entirely, leaving:

k = 8192/1365

Darksatanic 20:43, 23 October 2007 (BST)

Yes, but to a Duke fighting Phoenixes, they be priceless. --Mawenpea 14:27, 9 December 2007 (GMT)