What's the least skill levels that will result in character level 81?

Both of my level 81 characters got to where they are with power-leveling via Oghma Infinium. As such, I really didn't stop much to notice the smaller steps in level progression along the way.

What I have noticed now however is that, with all skills leveled to 100, my character level appears to be exactly 81.5 - that is, the character is level 81 and half of the character xp bar is full.

So, I'm interested to know what is the lowest set of skill levels that will add up to a character level at or just barely above 81.0?

Solution 1:

You would need 100 in all skills and 89 in one, so 1484 skill levels ([100 - 25] + 5 * [100 - 20] + 12 * [100 - 15] - [100 - 89]).

What's that, you want an explanation?

The formula to level up is:

XP required to level up your character = Current level * 25 + 75

if you remember your math classes and that whole sum(1..n) = (n + 1) * n / 2 business, that means we need

(81 * 80 / 2) * 25 + 75 * 80 = 87,000

skill ranks to level 80 times (from 1 to 81). We start off with one skill at 25, five at 20 and the rest at 15. Leveling each skill to 100 gives us

(100 + starting level + 1) * (100 - starting level) / 2

points, which brings us to 4725, 4840 and 4930 skill levels, respectively. If you level up all skills except one fully, we're at 83155 (4930 * 11 + 4840 * 5 + 4725), so still 3845 skill levels short. So with that last skill, we can solve for the the final level required:

(x + 15 + 1) * (x - 15) / 2 = 3845
(x + 16) * (x - 15) = 7690
x^2 + x - 240 = 7690
x^2 + x - 7930 = 0
x = [-1 + sqrt(1 - 4 * -7930)] / 2
x = (178.10 - 1) / 2 = 88.6
x rounds up to 89

Note:
Using a major or minor skill here wouldn't affect the calculation:

(x + 25 + 1) * (x - 25) / 2 = 87000 - (4930 * 12 + 4840 * 5 + 4725 * 0)
(x + 20 + 1) * (x - 20) / 2 = 87000 - (4930 * 12 + 4840 * 4 + 4725 * 1)
(x + 15 + 1) * (x - 15) / 2 = 87000 - (4930 * 11 + 4840 * 5 + 4725 * 1)

Also, you can test it out with an online character creator, like this half-baked one I threw together.

Edit:
Semi-interestingly, if you want to know the maximum number of levels required, it's only one more than the minimum. It would involve 8 skills at 100 and 10 skills at 99, so 10 less than the maximum number of skill increases available versus 11.

Edit 2:
In case anyone's confused about why I'm ignoring major/minor skills or anything else that might have affected a character in Morrowind/Oblivion, there's a great answer on this topic. All skills are born equal. Leveling a major skill Sneak from 30 to 31 is the same as leveling a standard skill Sneak from 30 to 31, which is the same as leveling a minor skill Archery from 30 to 31. All of them give you 31 points of "skill level XP" or whatever we want to call it. The bottom line is that these mechanics are understood and the above answer should be true for all races and all skills.

Solution 2:

I went ahead and crunched the numbers in Excel myself, to see if my own methods would yield the same answer as Bryan. Initially, I was surprised to find it didn't, but near the end of documenting my work here I seem to have stumbled across my error in process. Still I hate to waste a good write-up, so here's my alternative method:

TL;DR: This is how I used Excel to come to the same conclusions Bryan drew with just some basic high school algebra.

To reach Character Level 81 by these calculations, you need 17 level 100 skills and 1 level 89 skill. This requires traversing a total of 1,484 skill levels.

Instead of going with the more complicated maths I've forgotten since high school, I decided to go with very basic addition, subtraction, multiplication, and division, and let Excel do most of the leg work. Unfortunately, that means I had to generate a lot of cells I won't be showing here. I will post a sample though, so you can see this part at work.

The column headers are on row 6 because I ended up putting other calculations in the rows above, so all of the important numbers would be easily reachable nearby. I'll be showing some of the cells in the hidden rows later.

In column A, under the header Skill Level, I created a series of cells to mark levels 1-100. As you can see, I ended up using these for both Skill and Character level markers.

Column B is the Character XP that is earned for leveling up a Basic skill to the given level. What I'm calling "Basic" here are the skills for which the character receives no racial bonus. Levels 1-15 are set at zero, because these skills are given to the player on character creation, resulting in no XP earned.

Column C is the same as B, but for Secondary skills. Secondary skills are those which the player receives a +5 racial bonus, so they don't start giving XP until level 21.

Column D is the same as B & C, but for the Primary skill. This is the skill for which a player receives a +10 bonus, and does not begin earning XP until level 26.

In Columns B-C, all of the zeroes are manually entered. The non-zero cells are set to be equal to the cell in the same row, on Column A.

The Character Level Requirement column lists the XP required to progress from the previous level to the level for that row. This is according to the formula given in UESP:

XP required to level up your character = Current level * 25 + 75

So, for example, the formula in F24 is F23*25+75 because leveling your character to level 18 requires you to go from 17, multiply by 25, then add 75. Level 1 (not shown) is a manually entered zero. The formula then is populated for levels 2-81.

All that being done, I now wanted to figure out how much total Character XP was needed to progress through levels 2-81. This was simply done by SUM(F7:F87), and turned up the same result Bryan had: 87,000.

Next, to quickly determine how I could reach that goal, I wanted to know how much XP would be earned by leveling each type (Basic, Secondary, Primary) of skill to level 100, and how much would be earned in total for each skill group. For reference, I also wanted to know (you'll see I did this the hard way, as there is an easier solution with simple subtraction) how many levels the character would actually traverse when maxing those skills.

Row 1 is the total XP earned by leveling one of each class of skill to 100. The columns line up with the columns in the other part of the spreadsheet: B is Basic, C is Secondary, D is Primary. The formula for B1 is =SUM(B7:B106), and is copied over to the other cells in the row.

Row 2 is the total XP earned by leveling all skills of each class to 100. There are 12 Basic skills, 5 Secondary, and 1 Primary. The formulas are a simple multiplication of the cell above, by the number of skills in that class. B2=B1*12 C2=C1*5 D2=D1

Row 3 is the number of levels traversed to max one skill in each class. The formulas are based on the number of non-zero XP Earned cells in that column. For B3, it is =COUNTIF(B7:B106,"<>0"). This is copied over to the other cells in that row. This is the step where I could have gone with simple subtraction (100 - Number of non-earned levels).

Row 4 is the number of levels traversed while maxing all skills of each class. The formulas are similar to Row 2. B4=B3*12 C4=C3*5 D4=D3

Row 5 is really more for my amusement than anything, I guess. In the end, it makes little difference. It is just the amount of total XP from maxing one skill, divided by the number of levels traversed to accomplish the task. (B5=B1/B3 and so on)

All this is great and all, but how do we figure out how to get to Character Level 81? Since the Primary and Secondary skills seemed to have the highest relative value, I decided to start there. The following sheet portion might seem like a bit of organized chaos, but I hope I can explain it clearly. All formula references to cells not displayed below point to cells shown or described above.

H2 is the total of all the XP gained by leveling all of the player's Primary and Secondary skills. (H2=D2+C2) I2 is the amount of XP still required to hit Character Level 81 after all Primary and Secondary skills are at level 100. (I2=F2-H2) Now, I needed to get a quick idea of how many Basic levels would be needed to meet that remainder. (J2=I2/B1) So, that's all of 11 Basic levels plus most of the last one.

But how much of the last one? H3 tells us the value of 11 Basic levels. (H3=B1*11) H4 lets us know our current running total. (H4=SUM(H2:H3)) And, I4 tells us how much further we need to go. (I4=F2-H4)

So, with all except one skill leveled up to 100, we still have 3,845 XP left to go. I couldn't think of a simple way to have Excel automagically give me the number of levels I needed, so I went to a slightly more manual approach. I clicked on B7 (the first cell in the column representing XP earned per Basic level), and dragged my selection down the column until the Sum in the status bar exceeded 3,845. This occurred at level 89, with a total of 3,885.

So, we have 17 skills leveled to 100 and one leveled to 85. Using data from the above cells, we get 1,484 total levels required with the formula =(D4+C4+(11*B3)+(COUNTIF(B7:B95,"<>0"))). (First value added is total level-ups for Primary. Second is total level-ups for all Secondary skills. Third is total level ups for one Basic skill, multiplied by 11. Last is a count of the level-ups required to take one Basic skill up to 89.

Thanks to Bryan for the great lesson learned here: Remembering your High School algebra can save you a lot of Excel headache.