Daniel Heinlein

In the video game Path of Exile, it is possible to six-link an item. In the remainder, I abbreviate some expressions:

OoF	Orb of Fusing
VO	Vaal Orb
TOoF	Tainted Orb of Fusing

There are multiple ways to six-link an item:

Standard way

Apply OoFs until the item has six links.

Crafting bench

Pay 1500 OoFs on the crafting bench.

5 to 6

Pay 150 OoFs on the crafting bench to five-link the item. Then corrupt the item (using e.g. a VO or a Craicic Vassal with beast crafting to get 30% quality; note that the higher the quality of the item, the higher the probabilities to get more links while using OoFs or TOoFs). Then repeat: Apply a TOoF, if it is six-linked then done, else use the crafting bench five-link for 150 OoFs + 150 VOs (as the item is corrupted, crafting bench linking requires the same amount of VOs as OoFs).

4 to 6

Pay 5 OoFs on the crafting bench to four-link the item. Then corrupt the item. Then repeat until the item is six-linked: Apply TOoFs, whenever the largest group of linked sockets is three, apply the crafting bench four-link craft for 5 OoFs + 5 VOs. (The general remarks of the method 5 to 6 apply.)

These expected values and standard deviations (noted as ev or ev ± std) are computed via a suitable geometric distribution and a Markov chain, cf. the corresponding paper.

	OoF	VO	TOoF	Expected Cost
Price
Crafting bench	1500	0	0	300.00
5 to 6	300±212.13	150±212.13	2±1.41	175±176.78
4 to 6	15	10	6	128