User:Kim

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I'm Kim. I'm on Mari. I'm in Infernal.

<math>E_{c,t} = 1 + s_c E_{c+1,t} + f_c E_{c-1,t}</math>

The above is the formula that lets you determine how many upgrade stones you can expect to use to get your weapon to a certain level. <math>E_{c,t}</math> is the number of stones you should expect to use to get your weapon from its current level <math>c</math> to its target level <math>t</math>. <math>s_c</math> is the success rate at the current level -- it's 1 for level 0, 0.5 for levels 1 and 2, and 0.45 for levels 3 and 4. <math>f_c</math> is the failure rate; <math>s_c + f_c = 1</math>, of course. <math>c = t</math> implies <math>E_{c,t} = 0</math>.

You can make a system of equations out of those formulas and then solve them. The original matrix:

<math>\left( \begin{array}{cccccc} 1 & -1 & 0 & 0 & 0 & 1 \\ -0.5 & 1 & -0.5 & 0 & 0 & 1 \\ 0 & -0.5 & 1 & -0.5 & 0 & 1 \\ 0 & 0 & -0.55 & 1 & -0.45 & 1 \\ 0 & 0 & 0 & -0.55 & 1 & 1 \end{array} \right)</math>

The results:

Target Upgrade Level	1	2	3	4	5
Expected stones used from level 0	1	4	9	17.33	29.74
Expected stones used from level 1		3	8	16.33	28.74
Expected stones used from level 2			5	13.33	25.74
Expected stones used from level 3				8.33	20.74
Expected stones used from level 4					12.41