Blood Mage

Arcane Transfusion

At 2nd level, you have learned to draw upon the arcane that dwells within your blood. As a bonus action, you draw out the magic from your own blood and restore a spell slot of your choice. You take damage to your maximum hit points equal to 5 * spellslot level + half your wizard level. Any reduction in your maximum hit points caused by any of your blood mage abilities lasts until you finish a short or long rest. 

Blood Sacrifice

Starting at 2nd level, you sacrifice some of your health to mend another creature's injuries within 30ft. You take damage to your maximum hit points up three times your Intelligence Modifier and grant half that amount to a creature of your choice in range.

Blood Infusion

Starting at 6th level, you can transfer your blood into the spells you cast. Whenever you cast a spell, you can infuse it with your blood by reducing your maximum hit points by half your Intelligence Modifier (round down min. 1) and add twice the amount to either your attack roll or damage roll to the that spell. 

Blood Clot

Beginning at 10th level, the blood you channel helps ward off harm. While your maximum hit points are reduced, you have a +2 bonus to AC and are resistant to necrotic damage.

Bound In Blood

At 14th level, whenever you receive healing from a source you don't control, it now also heals your maximum hit points. Also, your Blood Infusion can now be used when you make a Saving Throw adding it to your roll. Additionally, your Blood Sacrifice goes up to five times instead of three.

Mathemagician 

Mathematical Studies 

When you choose this subclass at 2nd level, you become proficient in Investigation and you can make Investigation checks that use numbers or require calculations as a Bonus Action.

Mathematical Analysis 

Starting at 2nd level, you gain the ability to analyze and calculate probabilities, allowing you to gain insights into your opponents and the battlefield. You can use an Action to make an Intelligence (Investigation) check contested by their Deception check. On a failure, you cannot use it on this creature again for 24 hours. On a success, you gain one of the following insights:

• You learn the number and die level of hit die of a creature within 60 feet of you.
• You learn the AC of a creature within 60 feet of you.
• You learn the saving throw bonuses of a creature within 60 feet of you.

You can use this feature a number of times equal to your Intelligence modifier (minimum of once), and you regain all uses of this feature after finishing a long rest.

Geometric Patterns  

At 6th level, you have unraveled the ability to change your Area of Effect spells. When you cast a spell that targets an area with a radius of 5ft or more, you can apply your Geometric Patterns to it instead. You choose a number of 5ft cubes equal to your Intelligence Modifier + twice the spells level (min. 3). These become the new area of effects for your spell. They must still stay within the spells range, each one must connect to at least one other one horizontally or vertically and be within your line of sight. All other rules of the spell happen as normal. If the area of effect can be moved, it all moves in the same pattern as it normally would.

Equation Casting

At 10th level, you can use complex equations to cast spells. When you prepare your spells each day, you can choose to prepare a number of equation spells equal to your Intelligence modifier. These spells do not count against your total number of prepared spells. You can only prepare these spells for spell levels of which you can cast. Equation spells can be found at the end of this class.

Probability Manipulation 

At 14th level, you gain the ability to manipulate probability. As a Reaction, you can roll a d20. You can choose to add or subtract a number from the roll equal to your Intelligence modifier + proficiency bonus. This ability can be used to influence any d20 roll made within 60 feet of you, including ability checks, attack rolls, and saving throws. The target must be willing or unaware of your manipulation. This ability cannot be used to influence rolls made by deities or other entities with divine power. When you do this, you can choose to use your roll or allow the targeted creature to make their own roll.

You can do this a number of times equal to your Intelligence modifier. You regain all expended uses on a Long Rest.