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

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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?

Best Answer

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.

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