Geocaching: Using Multi-Billion Dollar Technology (and Math) to find Tupperware in the Woods CMC3...

Geocaching: Using Multi-BillionDollar Technology (and Math) to

findTupperware in the Woods

CMC3 Recreational ConferenceBruce 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 5040.

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 SegmentDistance (miles)




AB 0.5 BC 0.2 CE 0.8 DH 0.5

AD 0.5 BD 0.7 CF 0.6 EF 0.2

AC 1.1 BE 0.7 CG 0.6 EG 0.3

AE 1.1 BF 0.5 CH 0.3 EH 0.5

AF 0.9 BG 0.5 DE 0.1 FG 0.2

AG 1.0 BH 0.4 DF 0.2 FH 0.4

AH 0.8 CD 0.8 DG 0.2 GH 0.3

Build a Minimal Spanning Tree





AB 0.5 BC 0.2 CE 0.8 DH 0.5

AC 0.5 BD 0.7 CF 0.6 EF 0.2

AD 1.1 BE 0.7 CG 0.6 EG 0.3

AE 1.1 BF 0.5 CH 0.3 EH 0.5

AF 0.9 BG 0.5 DE 0.1 FG 0.2

AG 1.0 BH 0.4 DF 0.2 FH 0.4

AH 0.8 CD 0.8 DG 0.2 GH 0.3

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





AB 0.5 BC 0.2 CE 0.8 DH 0.5

AD 0.5 BD 0.7 CF 0.6 EF 0.2

AC 1.1 BE 0.7 CG 0.6 EG 0.3

AE 1.1 BF 0.5 CH 0.3 EH 0.5

AF 0.9 BG 0.5 DE 0.1 FG 0.2

AG 1.0 BH 0.4 DF 0.2 FH 0.4

AH 0.8 CD 0.8 DG 0.2 GH 0.3






AB 0.5 BC 0.2 CE 0.8 DH 0.5

AC 0.5 BD 0.7 CF 0.6 EF 0.2

AD 1.1 BE 0.7 CG 0.6 EG 0.3

AE 1.1 BF 0.5 CH 0.3 EH 0.5

AF 0.9 BG 0.5 DE 0.1 FG 0.2

AG 1.0 BH 0.4 DF 0.2 FH 0.4

AH 0.8 CD 0.8 DG 0.2 GH 0.3

The line segments chosen above give a different minimal spanning tree.






AB 0.5 BC 0.2 CE 0.8 DH 0.5

AC 0.5 BD 0.7 CF 0.6 EF 0.2

AD 1.1 BE 0.7 CG 0.6 EG 0.3

AE 1.1 BF 0.5 CH 0.3 EH 0.5

AF 0.9 BG 0.5 DE 0.1 FG 0.2

AG 1.0 BH 0.4 DF 0.2 FH 0.4

AH 0.8 CD 0.8 DG 0.2 GH 0.3

The line segments chosen above give yet a third minimal spanning tree.

More Spanning Trees

Minimal Spanning Tree #2 Minimal 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 #1 Minimal 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 miles Path of length 2.6 miles

The Real World

Theory vs. Reality

Theoretical Best Path Actual 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 [email protected] 530-541-4660 x314

Geocaching: Using Multi-Billion Dollar Technology (and Math) to find Tupperware in the Woods CMC3...

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