Geocaching Geometry Puzzle

I am looking for help with a math puzzle. The answer is the final coordinates to a geocache.

I have put in quite a bit of time and lots of graph paper trying to solve this one. I have tried programs and even bought the correct tools to work on this one. Still with the tools and the basic knowledge needed to work on this I have found I am no math wiz. I do spend quite a bit of time and learn everything I can before asking for help on these but some puzzles are beyond my skill level.

What I do understand is this is a method of surveying called Metes and Bounds. I have found that somewhere in the puzzle is missing info leaving you with trying to solve the missing distance and direction.

The cache is NOT located at the Point Of Beginning (POB) coordinates, but is in the same general area.

1. The POB in the puzzle is N 49° 13.314 W 123° 01.073
2. Beginning at the POB distant thereon North 60° 00' East 150.00 metres;
3. Thence North 30° 00' West 60.00 metres;
4. Thence North 60° 00' East, 90.00 metres to the beginning of a tangent curve concave Southeasterly and having a radius of 125.00 metres;
5. Thence Northeasterly along said curve through a central angle of 60° 00' an arc distance of 130.90 metres;
6. Thence tangent to said curve South 60° 00' East 70.00 metres to the beginning of a tangent curve concave Northeasterly and having a radius of 200.00 metres;
7. Thence Southeasterly along said curve through a central angle of 55° 00' an arc distance of 192.00 metres to the beginning of a compund curve concave Northwesterly and having a radius of 310.00 metres;
8. Thence Northeasterly along said curve through a central angle of 120° 00' an arc distance of 649.26 metres to the beginning of a compund curve concave Southwesterly and having a radius of 125.00 metres;
9. Thence Northwesterly along said curve through a central angle of 55° 00' an arc distance of 120.00 metres;
10. Thence tangent to said curve South 70° 00' West an indeterminate distance;
11. Thence North 60° 00' West 90.00 metres; said point being coincident with the true position of the geocache;
12. Thence North 90° 00' West 140.00 metres to the beginning of a tangent curve concave Southwesterly and having a radius of 160.00 metres;
13. Thence Southwesterly along said curve through a central angle of 40° 00' an arc distance of 111.70 metres to the beginning of a reverse curve concave Northwesterly and having a radius of 160.00 metres;
14. Thence Southwesterly along said reverse curve through a central angle of 30° 00' an arc distance of 83.78 metres;
15. Thence tangent to said curve South 80° 00' West 125.00 metres to the beginning of a tangent curve concave Southeasterly and having a radius of 125.00 metres;
16. Thence Southwesterly along said curve through a central angle of 115° 00' an arc distance of 250.89 metres;
17. Thence tangent to said curve South 35° 00' East an indeterminate distance;
18. Thence South 62° 55' East 191.78 metres to the true Point of Beginning, more or less.
• By the way, your diagram takes you out of the park and onto city blocks where there are buildings. I'm pretty sure this isn't a real metes-and-bounds description, just something to give you a lot of headache in calculating distances.
– user88
Commented Feb 25, 2015 at 8:05
• @JoeZ. - Just noticed your comment. Unless you are thinking something different, the geocache is still located in Central Park (I have added a map to my answer). This answer seems reasonable although the metes-and-bounds description could be fictitious.
– Len
Commented Mar 7, 2015 at 21:25
• Yeah, the answer to the cache is still in Central Park.
– user88
Commented Mar 7, 2015 at 21:26

There are two indeterminate distances in the metes-and-bounds description, each at a different bearing. This creates a vector space of possible endpoints of the path, only one point of which returns the path to the true point of beginning, which is (most likely) the pair of indeterminate distances you'll want in order to find the correct position of your geocache.

I've drawn a not-exactly-to-scale diagram of the surveying path to help you verify the directions, included below:

The flag is where the geocache is.

I'll try and follow you through the coordinates after every step, assuming that $49° 13.314' N$, $123° 01.073' W$ is the origin $(0, 0)$, and coordinates are over and up in metres, to help you figure out which pair of distances you want.

Beginning at the POB distant thereon North 60° 00' East 150.00 metres;

Travel 150 metres at a bearing of 60° to $(129.90, 75.00)$.

Thence North 30° 00' West 60.00 metres;

Then make a left turn and travel 60 metres, a vector of $(-30, 51.96)$, to $(99.90, 126.96)$.

Thence North 60° 00' East, 90.00 metres to the beginning of a tangent curve concave Southeasterly and having a radius of 125.00 metres;

Then turn right again and travel 90 metres, a vector of $(77.94, 45)$, to $(177.84, 171.96)$. Here you will arrive on the border of an imaginary circle with a radius of 125 metres and with the center southeast (technically, a bearing of 150°) of where you currently are.

Thence Northeasterly along said curve through a central angle of 60° 00' an arc distance of 130.90 metres;

Travel along the circle an arc length of 60°, changing your bearing from 60° to 120° as you do so. Since the radius is 125 metres, the total distance travelled will be $125 \times (60\pi / 180) = 130.90$ metres. However, the place you will end up is actually 125 metres due east, because the start point, end point, and center of the circle form an equilateral triangle.

So you'll end up travelling an effective vector of $(125, 0)$ to $(302.84, 171.96)$.

Thence tangent to said curve South 60° 00' East 70.00 metres to the beginning of a tangent curve concave Northeasterly and having a radius of 200.00 metres;

Assuming you've travelled 125 metres due east, you'll be turning 30 degrees right here. If you'd gone along the circular arc, you will have naturally ended up on the 120° bearing already. Walk 70 metres straight, a vector of $(60.62, -35)$, to $(363.46, 136.96)$.

Here we come across our second circular curve. This one has a center that is 200 metres northeast (at a bearing of 30°).

Thence Southeasterly along said curve through a central angle of 55° 00' an arc distance of 192.00 metres to the beginning of a compund curve concave Northwesterly and having a radius of 310.00 metres;

The calculation of the final position is a bit harder this time. The bearing changes 27.5° left to 92.5°, but the effective distance is calculated as $400 \text{m} \times \sin 27.5° = 184.7 \text{m}$, since it's not as simple as an equilateral triangle like the first time. This translates to a vector of $(184.52, -8.06)$, which means you'll end up at $(547.98, 128.90)$.

Once you've walked along this arc, you'll start walking along another arc immediately afterwards, which is why the description mentions a compound curve. (A reverse curve is an arc where you end up turning in the opposite bearing of the one you were just turning in).

Thence Northeasterly along said curve through a central angle of 120° 00' an arc distance of 649.26 metres to the beginning of a compund curve concave Southwesterly and having a radius of 125.00 metres;

This is equivalent to travelling 536.94 metres at a bearing of 5°, or a vector of $(46.80, 534.89)$, ending up at $(594.78, 663.79)$.

Thence Northwesterly along said curve through a central angle of 55° 00' an arc distance of 120.00 metres;

This is the final instruction until the first mystery distance. Here we end up travelling a distance of 115.44 metres at a bearing of -82.5°, or a vector of $(-114.45, 15.07)$, ending up at $(480.33, 678.86)$.

For the purposes of figuring out the mystery distances, you can treat all these instructions as just one giant way of saying "travel 480.33 metres east and 678.86 metres north".

Similarly, the second set of instructions until the starting point again, not including the mystery distance, makes you travel a grand total of 428.88 metres west and 331.24 metres south.

So, the total of all non-mystery instructions is 53.45 metres east and 317.48 metres north, and travelling some distance south 70° west (250° bearing) and then travelling some more distance south 35° east (145° bearing) will get you back to the starting point.

Suppose these two distances are $a$ and $b$ – then the following system of equations will let you figure out the correct distances:

\begin{align} a \sin 250^\circ + b \sin 145^\circ + 53.45 &= 0\\ a \cos 250^\circ + b \cos 145^\circ + 317.48 &= 0\\ \end{align}

Once you have this, you can plug in the first distance $a$ into the first "indeterminate distance" instruction and then walk 90 metres at a bearing of 300° to your geocache.

• All this effort deserves more than 4 upvotes! +1 :-) Commented Mar 7, 2015 at 22:41

Here is a supplement to the excellent analysis by Joe Z:

The two unknown distances are shown on the following chart (error corrected from previous chart). Relative to the start point, the geocache is 167.43 meters east and 638.34 meters north. This translates to N 49°13.658'and W 123°00.935'.

Here is the spreadsheet if you want to use it but Joe's analysis is much easier to follow.

• So I think the answers were intended to be 250 and 320 metres exactly, but due to angle rounding they came out differently than expected.
– user88
Commented Feb 24, 2015 at 5:14
• Thanks! Now to sit and go through this and figure it out and learn step by step. I can say I was way off on my math by looking at this. Commented Feb 24, 2015 at 10:36
• Out of curiosity, what did you try, and what did you come up with?
– user88
Commented Mar 7, 2015 at 21:28