Cache is NOT at posted coordinates- don't go there. Part of a series I'm working on, Things I Learned in School (TILIS). This will document various unique things I learned about throughout my years of schooling. For each cache, you'll have to master the concept to get the final coordinates.
Linear Programming
George Bernard Dantzig (1914-2005) was an American mathematical scientist, best known for his work in operations research, computer science, statistics, and economics. His most famous contribution was the simplex algorithm, which is used to solve linear programming problems, or a mathematical optimization problem with a set of linear constraints. With a given set of values and constraints, one can calculate an optimal solution to a problem. For example, if a bison tube manufacturer has 4 styles of bison tubes to sell at different prices, and a limited number of customers for each, she can figure out the optimal number of each bison tube to produce to maximize her profits. Linear programming is a method to solve this type of problem.
Simplex Method & Example
Although some simpler, 2-variable problems can be solved by graphing the constraints, the simplex algorithm allows us to solve any problem with any number of variables, be it 2 variables and 2 constraints, or 500 variables and 312 constraints. Simplex is too dense to explain on a geocaching page without prior knowledge, so we'll work through this example graphically. In our example, let's maximize the value of Z, where Z=3x+2y. However, x and y have some constraints:
2x + y ≤ 18 (the sum of 2*x plus y must be 18 or smaller)
2x + 3y ≤ 42 (the sum of 2*x plus 3*y must be 42 or smaller)
3x + y ≤ 24 (the sum of 3*x plus y must be 24 or smaller)
x ≥ 0 , y ≥ 0 (meaning both x and y are non-negative, either 0 or larger)
We'll graph our constraints onto a graph:
With linear programming, the optimal result will be at an intersection of lines (also known as a vertex). The yellow area in the graph are all x and y values that meet all our constraints. We'll calculate the Z value for each of the 5 vertices on the edge of our qualifying yellow area:
- A(0,14): 3(0)+2(14)=28
- B(3,12): 3(3)+2(12)=33
- C(6,6): 3(6)+2(6)=30
- D(8,0): 3(8)+2(0)=24
- E(0,0): 3(0)+2(0)=0
Our optimal solution is 33, with x=3, y=12. Now you try with this word problem:
AngelusCowl is looking to maximize his number of GeoPoints! For GeoPoints, each letterbox is worth 3 points, and each wherigo is worth 7 points. However, AngelusCowl has limited time and resources. Cowl has 230.368 minutes to find all his caches- each letterbox takes 3 minutes to find, and each wherigo takes 4 minutes. Cowl's car can travel up to 228.845 miles- each letterbox requires 2 miles of driving, and each wherigo takes 5 miles of driving. Finally, AngelusCowl's flashlight has 80.136 Watts of power- each letterbox needs 1.25 watts to find, and each wherigo need takes 1 watt of power to find. How many letterboxes and wherigos should AngelusCowl find to maximize his GeoPoints?
A variation of linear programming called integer programming would make sense here- results that are only whole, non-fractional numbers. However, for the sake of the puzzle and final coordinates, assume that AngelusCowl can find fractional geocaches.
The final should be accessible 24/7, but please use stealth with muggles.