Geocaching: Using Multi-Billion Dollar Technology (and Math) to find Tupperware in the Woods CMC3 Recreational Conference Bruce Armbrust- Lake Tahoe Community College April 28, 2012

What is Geocaching? Containers are hidden around the world. The coordinates of their locations are posted online at geocaching.com Users with hand held GPS receivers go to the posted coordinates. They search (and search and search and…) They open the container, sign the log sheet, trade items if desired, and move on to their next target.

What is it really? An excuse to explore the world. A chance to solve puzzles. A great way to spend free time. An OBSESSION!!!

The Math Behind The Machine

GPS Satellites

The Calculations Each satellite sends its position and time encoded within every signal. The receiver determines the change in time between transmission and receipt of signal. The location of the receiver is then on the sphere given by

The Calculations-continued In a perfect world, four satellites would be sufficient (or three with an assumption). In OUR world, the receiver’s clock may be out of time with the satellites and therefore an error is quite likely. This means the receiver is actually on the sphere

Return of The Calculations Expanding and rearranging terms gives Creating the right inner product turns this into Which allows us to form a matrix equation which has a least squares solution given more than 4 satellites. Even better, the solution to the least squares comes from a quadratic equation when enough “trickery” is applied.

Wandering Around The Woods

Traveling Salesman Problem A Salesman must visit each of n cities that are connected by a series of roads. Because of the well known identity Time=Money, she wants to do it as quickly as possible. What route should she take so that each city is visited and the distance traveled is minimized?

Wandering Geocacher Problem Mathprofessor needs to feed his geocaching addiction by finding n caches. Because he feels bad for turning his wife into a geo-widow, he wants to find them as quickly as possible. What route should he take so that each GZ is visited and the distance traveled is minimized?

My Prey…

Only 40,320 Choices

How Will We Find the Route? By picking a starting point, the number of possible routes drops to Obviously, we can use a computer to check all 5040, but where is the fun in that? Let’s try the Twice-round-the-Tree Algorithm

Twice-Round-the-Tree Start by creating a minimal spanning tree for the graph. Duplicate each edge of the tree and find a closed Eulerian path. Start at the point of your choice and follow the path. Skip any previously visited vertices and instead go to the next unvisited vertex on the path.

The Numbers Line Segment Distance (miles) Line Segment Distance (miles) Line Segment Distance (miles) Line Segment Distance (miles) AB0.5BC0.2CE0.8DH0.5 AD0.5BD0.7CF0.6EF0.2 AC1.1BE0.7CG0.6EG0.3 AE1.1BF0.5CH0.3EH0.5 AF0.9BG0.5DE0.1FG0.2 AG1.0BH0.4DF0.2FH0.4 AH0.8CD0.8DG0.2GH0.3

Build a Minimal Spanning Tree Line Segment Distance (miles) Line Segment Distance (miles) Line Segment Distance (miles) Line Segment Distance (miles) AB0.5BC0.2CE0.8DH0.5 AC0.5BD0.7CF0.6EF0.2 AD1.1BE0.7CG0.6EG0.3 AE1.1BF0.5CH0.3EH0.5 AF0.9BG0.5DE0.1FG0.2 AG1.0BH0.4DF0.2FH0.4 AH0.8CD0.8DG0.2GH0.3 The line segments chosen above give a minimal spanning tree.

Minimal Spanning Tree #1

Doubled Tree #1

Notes on Tree #1 This Eulerian path leads to four different routes. Not all of them have the same total length The shortest path (HCABGFEDH) has a total distance of 2.8 miles.

Best Path From Tree #1

The Numbers Line Segment Distance (miles) Line Segment Distance (miles) Line Segment Distance (miles) Line Segment Distance (miles) AB0.5BC0.2CE0.8DH0.5 AD0.5BD0.7CF0.6EF0.2 AC1.1BE0.7CG0.6EG0.3 AE1.1BF0.5CH0.3EH0.5 AF0.9BG0.5DE0.1FG0.2 AG1.0BH0.4DF0.2FH0.4 AH0.8CD0.8DG0.2GH0.3

Build a Minimal Spanning Tree Line Segment Distance (miles) Line Segment Distance (miles) Line Segment Distance (miles) Line Segment Distance (miles) AB0.5BC0.2CE0.8DH0.5 AC0.5BD0.7CF0.6EF0.2 AD1.1BE0.7CG0.6EG0.3 AE1.1BF0.5CH0.3EH0.5 AF0.9BG0.5DE0.1FG0.2 AG1.0BH0.4DF0.2FH0.4 AH0.8CD0.8DG0.2GH0.3 The line segments chosen above give a different minimal spanning tree.

Build a Minimal Spanning Tree Line Segment Distance (miles) Line Segment Distance (miles) Line Segment Distance (miles) Line Segment Distance (miles) AB0.5BC0.2CE0.8DH0.5 AC0.5BD0.7CF0.6EF0.2 AD1.1BE0.7CG0.6EG0.3 AE1.1BF0.5CH0.3EH0.5 AF0.9BG0.5DE0.1FG0.2 AG1.0BH0.4DF0.2FH0.4 AH0.8CD0.8DG0.2GH0.3 The line segments chosen above give yet a third minimal spanning tree.

More Spanning Trees Minimal Spanning Tree #2Minimal Spanning Tree #3

Hmmmm…. The shortest path for tree #2 is 2.8 miles. The shortest path for tree #3 is 2.9 miles. The shortest path (HCABGFEDH) has a total distance of 2.8 miles.

Best Paths Minimal Spanning Tree #1Minimal Spanning Tree #3

More questions Are these paths the shortest we can find? If not, how can we find a shorter one? What about real world implications?

Two better paths Path of length 2.7 milesPath of length 2.6 miles

The Real World

Theory vs. Reality Theoretical Best PathActual Route Traveled

Puzzle Caches

Some of my favorites A Cache Landing A Trying Triangle AP Calculus Sampler Calculus 301 Calculus 329 Canadians Have Large Clocks Crash Cache Sum Fun The Magic Square Cache Trig 106 Trig 110 Venn Diagram

The Ultimate FTF

Bruce’s Contact Information If you would like more information on Geocaching or the mathematics behind it, feel free to contact me. Bruce Armbrust a.k.a. mathprofessor Lake Tahoe Community College x314