Oath of Numeracy

In hidden alcoves, caves, and under cover of dark... Ancient Mathematicians practiced the numeric arts in secret. In their formulas, they believed that they were looking at the very essence of creation. Knights of the Oath of Numeracy continue the works of these great Mathematicians, using their faith in and knowledge of the great numerical arts to defend creation, extend the limits of human knowledge, and impart the knowledge of mathematics on the common man.

Tenets of Numeracy:

The tenets of the Oath of Numeracy focus on two things - preserving the beautiful universe that gives us the imperfect system known as Mathematics, and teaching this system to others. 

Protect this Beautiful System: Mathematics is the imperfect way that us mortals comprehend the perfect system that is creation. You must work to protect that system from harm.

Enlighten Others: Mathematics can be used to improve countless lives - but only if the masses knows how to use it. You must share your knowledge of Maths with any creature who wishes to become enlightened with this knowledge, and you must strive to find worthy students. 

Seek Further Knowledge: The knowledge mortals hold is incomplete. You must always be willing to learn more, and share it with others once you have achieved this knowledge. 

Oath Spells and Casting

Oath of Numeracy paladins can use their deeper understanding of Maths and the universe around them to twist the inherent magic around them into spells. However, doing so is a process that requires years of study, and intelligence.

You do not gain Oath spells in the same way other Paladin classes do. Instead, you may prepare spells from the Artificer Spell List as if they were paladin spells. These count as Paladin Spells for you. These spells must be prepared in the same way you would normally prepare spells from the paladin spell list, and count against the number of spells you can prepare each day.

As well, when you take this subclass at level 3, your Spellcasting Ability becomes Intelligence.

Spell Save DC = 8 + your proficiency bonus + your Intelligence modifier

Spell Attack Modifier = your proficiency bonus + your Intelligence modifier

Channel Divinity

When you take this oath at 3rd level, you gain the following two Channel Divinity options:

Quadratic Smite: You can use your Channel Divinity to alter the way that your weapon channels your Divine Smite. As a bonus action, you cause your Divine Smite to deal Force damage instead of Radiant for the next hour. During this hour, replace the damage that your Divine Smite does with the following equation: , where x is a randomly generated number between 0 and 17, inclusive. If you are using a spell slot that is higher than first level, the smite deals  extra damage for each spell level higher than first, where x is a randomly generated number between 0 and 17, inclusive. (A new random number, x, is chosen for each instance of extra damage.) This smite gains no extra damage against fiends or undead. On a critical hit, double the number of times you run each calculation.

 

Damage Division: You can use your Channel Divinity to alter reality, by dividing the result of harm. When a creature within 30 ft. of you would take damage, you can use your reaction to grant them resistance to all damage types present in the damage being taken, before damage is applied. This resistance lasts until the end of the target's next turn.

Calculated Strikes

Starting at level 3, when making a melee weapon attack while wearing medium or heavy armor, you can use your Intelligence modifier, instead of Strength or Dexterity modifier, for the attack and damage rolls.

The Law of Large Numbers and Probability

Starting at 7th level, the calculus of the universe starts to alter around you, reflecting theoretical probabilities instead of experimental.

When you would roll to determine the damage or hit points restored by an action, attack, spell, or feature, you instead calculate the average of rolling those dice and use that, rounding down.

As well, when a creature within 30 ft. of you rolls to determine the damage or hit points restored by an action, attack, spell or feature, you may change that roll to the calculated average of using those dice, rounded down. You may do this a number of times equal to your intelligence modifier, regaining all uses on a long rest.

 

This average can be affected by feats or skills that let you re-roll damage dice, such as "Great Weapon Fighting." Helpful link here: https://rpg.stackexchange.com/questions/47172/how-much-damage-does-great-weapon-fighting-add-on-average

 

As a helpful reminder, here is the average of some commonly rolled dice.

1d4 = 2.5

1d6 = 3.5

1d8 = 4.5

1d10 = 5.5

1d12 = 6.5

1d20 = 10.5

(So - if you were to attack with a Greatsword that does 2d6 +3 damage, instead of rolling dice, it would do 2 x 3.5 + 3 = 10 damage, with no dice rolled.)

Equal Outliers

Starting at Level 15, you have gained the ability to manipulate probability in your favor, so long as the other end of the outlier happens eventually.

As a reaction, when a creature within 60 ft. of you rolls a 15 or higher on a d20, or 5 or lower, you may change the number rolled on the d20 to a 10. If this causes an otherwise successful attack, saving throw, or ability check to fail, then that creature's next Attack, Saving Throw, or Ability Check is an automatic success. If this causes an otherwise failed Attack, Saving Throw, or Ability check to succeed, then that creatures next Attack, Saving Throw or Ability Check is an automatic failure. 

 Once you use this feature, you cannot use it again until you take a short or a long rest.

Having your π and Eulering it too.

At level 20, you have unlocked the ability to use your mathematical knowledge to extend the range of your effects, and enhance your own physical prowess.

As an action, enter a perfect state of mathematical focus. For the next minute, you gain the following benefits.
-The range of your Auras, Paladin Spells, and Oath of Numeracy Features are multiplied by Euler's Number, e, rounded up. (So 3)

- Add π (rounded down, so also 3) to your AC, as well as any Ability Check, Saving Throw, or Attack Roll you make.