The Math of Fusings

The goal of this thread is to determine the theoretical chances of rolling a max link item with an orb of a fusing.

Assumptions:

An orb of fusing has an equal chance of yielding any of the possible outcomes.

An orb of fusing will never yield the current configuration.

Contrary to their appearance sockets are actually in a line, and only 2 sockets next to each other may be linked.

2 Socket

o o, o-o

With only 2 options an orb of fusion will always give you the other option! Guaranteed results.

3 Socket

o o o, o o-o, o-o o, o-o-o.

4 possible configurations, so 3 possible results. 1/3 (33%) chance of getting a 3l.

4 Socket

o o o o, o o o-o, o o-o o, o-o o o, o-o o-o, o o-o-o, o-o-o o, o-o-o-o.

8 possible configurations, so a 1/7 (14%) chance of getting a 4 link.

Now if you're the mathing person you might have picked up a pattern. The number of possible scenarios 2^(S-1) with S being the number of sockets the item has. With the chance of getting a max link being 1/((2^(S-1)-1).

This yields 16 possible configurations, which gives a 1/15 (6.6%) chance of getting a 5 link.

This would yield 32 possible configurations, which gives a 1/31 (3.2%) chance of rolling a 6 link.

Now in my readings and experience is has taken a lot more than 15 fusings roll a 5 link. So either my first assumption is wrong, or my math or wrong, or both.

There is not equal chance to get any outcome I believe. There is a very very steep drop off for additional links, which is why they are so rare.

The fault is in the assumption.

This math has already been done, let me see if I can find a link.

edit: can't find the link, and a lot of what I said below is wrong:

First, the algorithm doesn't work by "picking" a pre-determined valid setup. It iterates on the links individually and gives each a particular chance (50%?) of fusing, so long as fusing that link would result in a valid configuration.

So each of your above "configurations" individually has a certain probability to appear. All the probabilities should add to 100%. The 2-D grid has a huge impact on these probabilities, so you can't flatten the picture.

It's on the order of 1 in 300 (500?) fusings to get 6link. Having trouble finding the link.

"

Saffell wrote:

Contrary to their appearance sockets are actually in a line, and only 2 sockets next to each other may be linked.

As Zakaluka pointed out, the orientation does matter. If you simplify the socket layout into a line you miss some possible configs. The text formatting makes drawing them annoying so I'll have to describe them:

No links (1)

1L = N/A (0)

2L, 2 unlinked (4)
* Rotate the link top, right, bottom, left

2x2L (2)
* Linked horizontally
* Linked vertically

3L, 1 unlinked (4)
* Rotate the unlinked socket to each position

4L
* Forms a "C" shape. Rotate the gap N, E, S, W


0L____ 1
1L____ 0
2L____ 4
2x2L__ 2
3L____ 4
4L____ 4

Total=15

"

FriarJon wrote:

"

Saffell wrote:

Contrary to their appearance sockets are actually in a line, and only 2 sockets next to each other may be linked.

As Zakaluka pointed out, the orientation does matter. If you simplify the socket layout into a line you miss some possible configs. The text formatting makes drawing them annoying so I'll have to describe them:

No links (1)

1L = N/A (0)

2L, 2 unlinked (4)
* Rotate the link top, right, bottom, left

2x2L (2)
* Linked horizontally
* Linked vertically

3L, 1 unlinked (4)
* Rotate the unlinked socket to each position

4L
* Forms a "C" shape. Rotate the gap N, E, S, W


0L____ 1
1L____ 0
2L____ 4
2x2L__ 2
3L____ 4
4L____ 4

Total=15

I challenge you to find me 1 single 4s item that has the left 2 sockets vertically linked. Or a 5s/6s item with those same 2 sockets linked.

"

Zakaluka wrote:

This math has already been done, let me see if I can find a link.

First, the algorithm doesn't work by "picking" a pre-determined valid setup. It iterates on the links individually and gives each a particular chance (50%?) of fusing, so long as fusing that link would result in a valid configuration.

So each of your above "configurations" individually has a certain probability to appear. All the probabilities should add to 100%. The 2-D grid has a huge impact on these probabilities, so you can't flatten the picture.

It's on the order of 1 in 300 (500?) fusings to get 6link. Having trouble finding the link.

I considered this as well, and it might be more true. This concept is a lot easier to discuss in binary train of though. However, the end result is that half that states arn't valid, i.e. a socket is set a fusable, but both other sockets next to it are unfused. This would result in a LOT of unlinked rolls.

I can demonstrate the math if someone really desires. But the equation for the number of states including mismatched stated is 2^S. Which would make the equation for chance to get a 4L 1/((2^S)-1)

This yields a 1/15 or 6.7% chance to roll a 4l from 4s. A 1/31 3.2% chance of rolling a 5l from 5s. And a 1/63 or 1.6% chance of rolling a 6l from a 6s.

This still falls well below the experience values.

Ah, he actually appears to be right on that point. Sorry!

But there's still something missing. We've been told that a 6-link has 1 in 300 or so odds. I actually can't find the dev post now, went looking for it, gave up after 20 mins.

Need a solid guess at how the fusing algorithm actually works to come up with probabilities. If it were just an equal 50% chance of each link being fused, odds would just be 1 in 2^5 or 1 in 32. This is clearly not correct. It also can't be equal odds of every possible combination appearing, as you've demonstrated.

"

Saffell wrote:

An orb of fusing has an equal chance of yielding any of the possible outcomes.

I don't believe this is true. I have nothing to back this up, but from experience, the chances are heavily weigthed towards less links than max links, so it's not equal chance for all configurations.

Basically, you're wasting your time trying to figure chances of max links happening, because you have no idea the chances GGG gave for each configurations.

The probability of rolling any specific link grouping is a number that GGG has carefully chosen. The only way to approximate these numbers is by calculating the mean and standard deviation over a large enough sample. There was a thread a while back that attempted to do this but ultimately GGG can shift these probabilities whenever they want based on the current state of the economy and gameplay.