How to find a Stronghold using two Eyes of Ender
WARNING:
This will require an advanced knowledge of 'Maths'.
If you do not like math, stop reading! Your head will hurt!
The method we are going to use to find a Stronghold is a method called Triangulation (Not the typical triangulation with angles, but with vector intersections).
How we find the stronghold will look a little bit like this:
If you still don't get it then imagine this:
When you throw an Eye of Ender from your position, imagine drawing an infinitely long line in the direction of the Eye.
You know that line eventually leads to the stronghold, but where?
Well what happens if you make a second line, but in some far distance away from your first throw?
The two lines will intersect at some point, which is exactly where the stronghold is!
Also, later in this guide you may start asking: "Where does the i and k come from?" This is ijk notation for unit vectors.
Simply put, any x values are followed by an i, y values by a j, and z values by a k.
(e.g. (23, 27, 42) = 23i + 27j + 42k)
But in this guide, the y value is irrelevant, so there will only be i's and k's.
First, pick any location and record the x and z coordinates to the nearest integer.
(-849, 4985)
Second, throw an Eye of Ender, then record the x and z coordinates directly beneath it to the nearest integer.
(-847, 4974)
Repeat steps 1 and 2 again at least 100 blocks away from your first location, that is not in the same direction as the Eye of Ender.
Know that the further away you take these measurements, the more accurate the result will be.
(-1181, 4955)
(-1178, 4944)
We have 4 coordinates--2 vectors.
Let line # be (Throwing position) -> (Eye of Ender position)
Let line 1 be (-849, 4985) -> (-847, 4974)
Let line 2 be (-1181, 4955) -> (-1178, 4944)
Now what? Here is where math comes in handy.
Taking the differences of the coordinates for each line we get:
(Eye of Ender position) - (Throwing position), but in this case I made a mistake and did it the other way around.
Don't worry! It will still yield the same result in the end--we just have to work with more negative numbers instead.
Line 1: (-849, 4985) - (-847, 4974) = -849i + 4985k - (-847i + 4974k) = -2i + 11k so the difference is (-2, 11)
Line 2: (-1181, 4955) - (-1178, 4944) = -1181i + 4955k - (-1178i + 4944k) = -3i + 11k so the difference is (-3, 11)
With those, we can make ourselves parametric equations of each line:
r = a + tb
Where r is the resultant position vector,
a is the throwing position (a position on the line),
b is the difference of the coordinates we just calculated (the directional vector),
and t is the number of Eye of Ender distances.
Line 1:
Line 2:
Now that we have our parametric equations
and
we can now calculate their intersections (when both x's and z's are the same)!
You can do this several ways, but I prefer to use the Elimination method.
For more information on the Elimination method, and other methods of solving linear systems or about linear systems in general, refer to this easy-to-understand and fun website: http://www.mathsisfun.com/algebra/systems-linear-equations.html
Now guess what? When verifying our results, we just finished finding the coordinates of the stronghold:
x = -169
z = 1243
So...
(-169, 1243)
is where we will find our stronghold!
*Disclaimers:
These may not be the exact coordinates of the stronghold due to the precision of our measurements.
Measuring from two locations that are too close together will increase the margin of error,
and because we rounded our coordinates, that also increased the margin of error.
But despite this margin of error, this technique is a very good approximation of where the stronghold is.
It will reduce the cost of using hundreds of Eyes of Enders to find strongholds, which is useful since enderman are quite uncommon.
Actual coordinates lead by the Eye of Ender:
(-283, 1156)
Actual End portal location:
(-311, 1191)
I saved quite a lot of Eyes of Ender--to start throwing and following them at (-169, 1243) rather than at (-1181, 4955), I saved a lot of resources!
Big Thanks to Mhayden for allowing me to use his End portal!
And another thanks to pikachu_201 for the idea!
WARNING:
This will require an advanced knowledge of 'Maths'.
If you do not like math, stop reading! Your head will hurt!
The method we are going to use to find a Stronghold is a method called Triangulation (Not the typical triangulation with angles, but with vector intersections).
How we find the stronghold will look a little bit like this:
If you still don't get it then imagine this:
When you throw an Eye of Ender from your position, imagine drawing an infinitely long line in the direction of the Eye.
You know that line eventually leads to the stronghold, but where?
Well what happens if you make a second line, but in some far distance away from your first throw?
The two lines will intersect at some point, which is exactly where the stronghold is!
Also, later in this guide you may start asking: "Where does the i and k come from?" This is ijk notation for unit vectors.
Simply put, any x values are followed by an i, y values by a j, and z values by a k.
(e.g. (23, 27, 42) = 23i + 27j + 42k)
But in this guide, the y value is irrelevant, so there will only be i's and k's.
First, pick any location and record the x and z coordinates to the nearest integer.
(-849, 4985)
Second, throw an Eye of Ender, then record the x and z coordinates directly beneath it to the nearest integer.
(-847, 4974)
Repeat steps 1 and 2 again at least 100 blocks away from your first location, that is not in the same direction as the Eye of Ender.
Know that the further away you take these measurements, the more accurate the result will be.
(-1181, 4955)
(-1178, 4944)
We have 4 coordinates--2 vectors.
Let line # be (Throwing position) -> (Eye of Ender position)
Let line 1 be (-849, 4985) -> (-847, 4974)
Let line 2 be (-1181, 4955) -> (-1178, 4944)
Now what? Here is where math comes in handy.
Taking the differences of the coordinates for each line we get:
(Eye of Ender position) - (Throwing position), but in this case I made a mistake and did it the other way around.
Don't worry! It will still yield the same result in the end--we just have to work with more negative numbers instead.
Line 1: (-849, 4985) - (-847, 4974) = -849i + 4985k - (-847i + 4974k) = -2i + 11k so the difference is (-2, 11)
Line 2: (-1181, 4955) - (-1178, 4944) = -1181i + 4955k - (-1178i + 4944k) = -3i + 11k so the difference is (-3, 11)
With those, we can make ourselves parametric equations of each line:
r = a + tb
Where r is the resultant position vector,
a is the throwing position (a position on the line),
b is the difference of the coordinates we just calculated (the directional vector),
and t is the number of Eye of Ender distances.
Line 1:
Code:
r = a + tb
(result) = (throwing pos.) + t(difference)
(x, z) = (-849, 4985) + t(-2, 11)
xi + zk = -849i + 4985k + t(-2i + 11k)
Simplifies to
xi + zk = (-849 - 2t)i + (4985 + 11t)k
Which can be reduced down to two equations that make our parametric line:
x = -849 - 2t
z = 4985 + 11tLine 2:
Code:
r = a + sb
*Change [I]t[/I] to [I]s[/I] to keep it separate from the other equation.*
(x, z) = (-1181, 4955) + s(-3, 11)
xi + zk = -1181i + 4955k + s(-3i + 11k)
Simplifies to
xi + zk = (-1181 - 3s)i + (4955 + 11s)k
Which can be reduced down to two equations that make our parametric line:
x = -1181 - 3s
z = 4955 + 11sNow that we have our parametric equations
Code:
x = -849 - 2t
z = 4985 + 11t
Code:
x = -1181 - 3s
z = 4955 + 11sYou can do this several ways, but I prefer to use the Elimination method.
For more information on the Elimination method, and other methods of solving linear systems or about linear systems in general, refer to this easy-to-understand and fun website: http://www.mathsisfun.com/algebra/systems-linear-equations.html
Code:
First, make the two [I]x[/I] equations equal to each other
-849 - 2t = -1181 - 3s
then bring both variables to the left side and the constants to the right, which simplifies to
-2t + 3s = -332
Now, do the same to the two [I]z[/I] equations!
4985 + 11t = 4955 + 11s
11t - 11s = -30
So, being left with
-2t + 3s = -332
11t - 11s = -30
We shall align them on top of one another
-2t + 3s = -332
11t - 11s = -30
From here, we can multiply the first equation by 11, multiply the second equation by 2, then add the two equations to eliminate [I]t[/I].
[-2t + 3s = -332] * 11 = [-22t + 33s = -3652]
[11t - 11s = -30] * 2 = [22t - 22s = -60]
[-22t + 33s = -3652]
+ [ 22t - 22s = -60 ]
======================
[ 0t + 11s = -3712]
Which makes
11s = -3712
s = -3712 / 11
s = -337.454545
Then substitute [I]s[/I] back into [I]-2t + 3s = -332[/I] (or either equation for that matter)
-2t + 3(-337.454545) = -332
Which makes
-2t = -332 - 3(-337.454545)
t = (-332 - 3(-337.454545)) / -2
t = -340.181817
Now to verify our results, we substitute both [I]t[/I] and [I]s[/I] into both equations and make sure the left side is equal to the right side.
x equations:
-849 - 2t = -1181 - 3s
-849 - 2(-340.181817) = -1181 - 3(-337.454545)
-168.636366 = -168.636365
(Round to nearest integer)
-169 = -169
z equations:
4985 + 11t = 4955 + 11s
4985 + 11(-340.181817) = 4955 + 11(-337.454545)
1243.00001 = 1243
(Round to nearest integer)
1243 = 1243Now guess what? When verifying our results, we just finished finding the coordinates of the stronghold:
x = -169
z = 1243
So...
(-169, 1243)
is where we will find our stronghold!
*Disclaimers:
These may not be the exact coordinates of the stronghold due to the precision of our measurements.
Measuring from two locations that are too close together will increase the margin of error,
and because we rounded our coordinates, that also increased the margin of error.
But despite this margin of error, this technique is a very good approximation of where the stronghold is.
It will reduce the cost of using hundreds of Eyes of Enders to find strongholds, which is useful since enderman are quite uncommon.
Actual coordinates lead by the Eye of Ender:
(-283, 1156)
Actual End portal location:
(-311, 1191)
I saved quite a lot of Eyes of Ender--to start throwing and following them at (-169, 1243) rather than at (-1181, 4955), I saved a lot of resources!
Big Thanks to Mhayden for allowing me to use his End portal!
And another thanks to pikachu_201 for the idea!
Last edited:
