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From the second equation we get that cheeseburger plus fries-squared is five. Subtracting cheeseburger, which is six, from both sides, we get that fries-squared is negative-one. Math fans will know that there are two solutions to this; either fries are the "imaginary unit" 𝒾 or they are its negative, -𝒾. We'll do the rest of the problem with 𝒾, keeping in mind that at the end we should also take the complex conjugates as solutions.

Finally, we have that cup to the power of fries, minus cup, equals three. Replacing fries with 𝒾, and moving a cup to the other side, we get that cup-to-the-𝒾 is equal to cup-plus-three.

Now, the weird part about this is the cup-to-the-i. The problem with this is that complex exponentiation is technically not a thing. That is to say, there is no one function which is mathematically equal to "input-to-the-power-of-𝒾". In fact, there are infinitely many such functions.

Fortunately, due to reasons that take about six pages to explain (trust me I've done it), there is one particular function that many people have agreed is "the most reasonable one". This is not a mathematical notion, but a human preference. Seeing as this question was presumably written by a human, I am comfortable with using this function.

So, what function is this? Well, given a complex number r∠θ written in polar form (if you don't know what that means don't worry), where -π < θ ≤ π, then (r∠θ)^𝒾 = e^(-θ)∠ln(r).

Applying this to our problem a value r∠θ will be a possible solution for cup if e^(-θ)∠ln(r) = r∠θ + 3. Splitting this into real and imaginary parts, we get two equations: e^(-θ) cos(ln(r)) = r cos(θ) + 3 and e^(-θ) sin(ln(r)) = r sin(θ). We can graph these equations on Desmos:

let the value of the glass = z, for z ∈ ℂ:

z^(±i) = 3+z

let z = r·e^(iθ) for r,θ ∈ ℝ, -π < θ ≤ π

∴ z = r·e^i(θ+2πn) for all n ∈ ℤ

∴ (r·e^i(θ+2πn))^(±i) = 3+r·e^i(θ+2πn)

distribute powers (apologies for the use of ∓):

r^(±i)·e^(∓(θ+2πn)) = 3+r·e^i(θ+2πn)

convert to the same base:

e^i(±ln(r))·e^(∓(θ+2πn)) = 3+r·e^i(θ+2πn)

split into real and imaginary components:

re: cos(±ln(r))·e^(∓(θ+2πn)) = 3+r·cos(θ)

im: sin(±ln(r))·e^(∓(θ+2πn)) = r·sin(θ)

Quantum witch who has figured out how to store her nightmares, intrusive thoughts, and crippling depression into patterns compatible with human neural impulses and can project them in the form of hexes onto other people, or reverse the process to siphon thoughts.

Guy who had his entire digestive tract cybernetically enhanced so it constitutes over 1/3 of his body mass but he can use all his organs as extra limbs or attack with them like a starfish throws its stomach at its prey.

Retired military robot whose brain is a daisy chain of forty rat brains who is still trying to get its personhood legally recognized and resents not being able to do anything for itself without the aid of a "service human" who legally speaking technically owns it as a pet.

Guy who has had his entire skeleton replaced with a network of low level AI centipedes allowing him to react at incredible speeds and move in ways that shouldn't be possible.

House sized anthill which lives underground and interacts through a series of dozens of drones which can temporarily assemble into a humanoid shape if needed.

Orca who had most of her body destroyed by a depth charge during one of many surface wars and had her head and remaining spine installed into a colossal multiarmed mech suit to be able to collect the disability compensation after she sued, and uses it to maintain a coffee bar and support group focused on civilians injured in military or police actions.

Guy who installed entropic conversion units at key joints of his body which allows him to become super fast, hyper intelligent, super strong, and indestructible for ten minutes, but anything within a ten foot sphere around him at the start gets turned to ash.

Ratbot has a real uphill battle because if it wins personhood the military has to start paying benefits to every functional ratbot and they were created as disposable soldiers with human like intelligence but without any regulations about their rights. As far as the military is concerned they just need to stall until ratbot parts all run out and the one with the lawsuit breaks down permanently. Bad news for ratbots.

The mosquito hybrid is a low income area doctor. Wealthy people can afford nanoplants that run off vat grown synthetic material and have near perfect cell regrowth and life extension. Mosquito hybrid found their nanoplant as a junk discard experimental model and needed a set amount of insect genetic material before it worked.

FINALLY FINISHED