"The Men who Wear the Star"

Good evening everypony, this week I bring unto you a logic puzzle of a man's ancestors and their military ranks. (Evan plus his four paternal ancestors)
1: "Martin is the grandson of the man who retired with one-star."
2: "Jacob is not Randolph's son." 
3: "Nathaniel had two more stars than Evan's great-grandfather
4: "Either Jacob or Martin had three-stars"
 From clue three, Evan's great-grandfather can have no more than two-stars, and Nathaniel (Nate) must have either three or four. However Nate is not Martin or Jacob, so he must have 4-stars and Evan's great grandfather must have two.
 From clue 1, Martin must be Evan's grandfather because the position of great-grandfather is filled with a two-star holder.
 Jacob and Randolph cannot be next to each other and so must be the father and G grandfather, or 2G grandfather, but this puzzle is unsolvable for that very reason. Nate must be the father because that is the position lacking a star and a name but that forces a violation of clue 2.

"Making sense, hah. What fun is there in making sense?" ~ Discord.

Linear Functions

In the story problem, a shoe factory has certain parameters for two types of soccer shoes.
1) each pair goes through a two-step process
2) Outdoor shoes (X) take 2hrs in 1st step: 1hr in the 2nd, yielding 20$
3) Indoor shoes (Y) take 1hr in step 1: 3hrs in step 2, yielding 15$
4) 40hrs are available for step 1
5) 60hrs available for step 2
Knowing this, the maximum number of either shoe that can be created is 20 pairs [40/2, 60/3] {X<= 20, Y<=20}   {X>=0,  Y>=0}    F(X,Y)= 20X+15Y
With this, the feasible region is a square defined at the points: (0,0); (0,20); (20,0); and (20,20)
Max F= (20,20); [20*20+(15*20)]=  700$

"Dumb fabric." ~Sweetie Belle

Week 2

*The solutions of inequalities are all points that satisfy the conditions either above or below the line, dependent upon the nature of the inequality. However, with an inequality system, the solution is the area of overlay betwixt the two(+) inequalities.
*{Y<4
  {X>-4 Solutions include: (-3.99,3.99)_(-3.99,-infinite)_(infinite,3.99)_(infinite,-infinite) and all area within that.