f to doubt

Tell us your bug herding secrets, op
Explain to us how point B isn't the solution

P2428
Here's a hint: Notice how your paths over the top are longer than the brown path.

You can figure out the solution by unwrapping the thing onto a flat plane: I can't be bothered to solve the resulting system of equations, so I'm going to guesstimate and say it's on the top face at coordinates (0.75,0.75,2) with A being at (0,0,0) and B being at (1,1,2).

P2429
What's the obvious trick to this? Other than calculating the minimum to origin along the gray line on the top face, is there a reason we should know the answer was the answer.
Instead of undergraduate (high school?) calculus I just did
(defun min-of (x y &aux (ox -0.5) (oy -2.5)) "
Imagining the origin in the middle of the 'far-away face'"
(let* ((x* (1+ y))
(y* (- x)))
;;(format t "point-in: ~a ~a~%" x y)
;;(format t "point-tr: ~a ~a~%" x* y*)
(flet ((norm (a b) (sqrt (+ (expt a 2) (expt b 2)))))
(min (norm (- x ox) (- y oy))
(norm (- x* ox) (- y* oy))))))

(defvar *nummax*
(sort
(loop for c-1 below 100 collecting
(list (min-of (- (/ c-1 100) 0.5)
(- (/ c-1 100) 0.5))
c-1 c-1))
'>=
:key 'first))

(print (car *nummax*))
(terpri)

;; I just don't remember this identity. Dau must be rolling in his grave

P2436
Sniped by not refreshing a page. What's the system of equations?
Or is it about like using calculus to find a maximum distance from origin along the gray line?

P2442
You can figure it out by sketching out what region you can reach by traveling x distance for various values of x. The system of equations he's talking about is probably the intersection of the boundary lines at which one type of route becomes preferable to another.

The system I was thinking about was:
>1 equation to constrain the point on the diagonal of the top face
>2 equations to express the distance from A to the point, one for the line passing through two faces and one for the line passing through three faces
And then you find the point that maximizes the result of those last two.

P2486
Alright what I'm hearing is that I overfocused on the word minimize from the problem. The maximum distance is where the values are equal, not that that really matters I guess.
So using the change of origin to the middle of the top face and parameterizing P2440 by t I need t such that
(1)
(t+0.5)^2+(t+2.5)^2=(t+1.5)^2+(t-2.5)^2
which is one equation and one unknown. Which 0.25 is a solution of. Expressed rationally I guess that's
(2)
(3/4)^2+(11/4)^2=(7/4)^2+(9/4)^2
From which I can drop the quarters then
(3)
3^2+11^2=7^2+9^2=130
which seems nice but aside from t conveniently being a number of quarters for (2), I can't see why t=0.25 is a special point on the shape in op. I guess instead of (1), I should be expressing t in terms of the dimensions of the box to get a general answer.

P2404

Interesting problem, thanks for posting.

My solution in picture.

Get in that locker, nerd!

P2660
Great writeup, I am now visually aware that y_1=x and y_2=3(1-x) intersect at 3/4.