Words are embedded, meaning they are associated with vectors. Then, the meaning of the statement A : B :: C : D is that D, relative to C is the same as B, relative to A. In vector language, this means our statement is D - C = B - A. Thus, the machine, when calculating relations between words A and B calculates A : B = B - A, and returns the word closest to this resulting relative displacement.
Notably, solvers can leave either of the inputs blank. The machine treats the blank input as the zero vector, so inputting blank : A returns A and inputting A : blank returns -A.
For example, to solve for A, in the first line, we simplify
man : woman :: A : herself
woman - man = herself - A
A = herself - (woman - man)
First, we enter into the machine (man : woman) to get “uh”, then enter (uh : herself) to get himself. Thus, A = himself.
Since the machine allows us find differences between words and allows you to invert words, we can solve for any arbitrary combination of words. Translating the given relationships to algebra and solving for the unknowns, we get the following key:
A = himself
B = spain
C = brazil
D = computer
E = medicaid
F = municipality
G = deer
H = hierarchy
I = ball
J = heel
The + sign indicates that we should concatenate J and I, giving our answer of HEELBALL.