As of 7 April 2010 one of the towers has been
dismantled and removed. This was the radio mast, at 38° 53.161,
175° 15.605. You may wish to pretend there is a large tower in this
position when attempting to crack this puzzle!
There are more towers than the three I chose to use for the
distance measurements, and those that were used are not necessarily
the tallest, or easiest to find, or easiest to reach. But they are
all within 1 km of the listed coordinates.
This is a more difficult derivative of the "first generation"
geometry type caches such as
The Geometry of Owhango (about 20km south of Taumarunui by road
if you want to warm up with that one first). What makes this one
more difficult is that you aren't told which distances go with
which clue, or exactly which clues to use.
Solving this puzzle is analogous in some ways to the problem a
GPS receiver has to solve to find its location. In that case the
receiver has to determine a very accurate time in order to work out
its distance from each satellite. Each GPS satellite carries its
own atomic clock, but each receiver has a relatively inaccurate
clock (it won't hold an accurate time over a long period). This
means the receiver's local time is not accurate enough to measure
the range from each satellite without some other technique, because
the calculated range (ignoring corrections for atmospheric effects)
is essentially obtained by multiplying the speed of light by the
time difference between when the signal was sent (encoded in the
signal) and when it was received. The "trick" performed by your
receiver to correct the initially inaccurate internal clock is
rather clever. In effect, it adjusts the time on that clock until
the calculated distances from a number of different satellites
generate an intersection at a single point - hopefully your
position! So, your GPS receiver is actually determining a very
exact time, and also the position simultaneously.
You can read more about this in Trimble's excellent description
of how GPS works, and
especially the section on getting perfect
timing.
In your case you'll need to determine which subset of towers to
use, and which distance to associate with each of those towers to
get a good intersection. When you've done that you will also have
found a possible location for the cache! If you associate the
distances mentioned above with the wrong towers, then you will
generally not find a feasible solution. On the other hand, you may
also find you get more than one solution – in which case you
will have to guess which is more likely, or perhaps even search
them all - all part of the challenge! I've written some software to
simulate the whole process, including allowing for realistic errors
in measured positions, and have used that to choose a set of clues
and a final hiding place that shouldn't produce too many false
solutions. But I'm not guaranteeing a single solution either - it
depends on the accuracy of both your and my measurements of the
various locations. I've double checked, and taken multiple readings
(roughly averaged) in order to help reduce any errors in mine.
Some of the towers are on private property or in places where
nearby hazards exist. So please be sensible and careful - get
permission to access them as required, or else use triangulation or
some other technique to fix the "exact" positions from a more
convenient nearby position. You can get quite close to all of them
without difficulty. But remember, the more accurately you can
determine those locations, the more accurate your final solution is
likely to be! When you have determined the location, you need to
make your final approach FROM THE WEST. There's a good spot to fill
in the log book and so on about 30 metres to the south east of the
actual hide. That spot is on private land but I've spoken to the
owners and they are comfortable about an occasional (well behaved)
geocaching visitor or two. :-)
As mentioned above, I set this up and tested it using a Python program that
searched all possible solutions in a brute force manner. The same
program can solve this class of puzzle from scratch given the
position of all the possible clues, and the distances to the cache.
But there are ways this type of puzzle can also be solved using a
more basic approach, for example with "paper and pencil" (and
perseverance probably!), especially if the number of potential
clues isn't too large. The "nearest waypoints" page on your GPS
receiver might even do the job if you can "intuit" the general area
where the cache might be hidden. Perhaps “proximity
rings” can be used also – but my basic yellow etrex
doesn’t have that feature so I can’t be
sure…
Please also try to remember to note the
distances your GPS reports from the cache to the three key towers,
and include those in your log entry so we can see how much
variation there is! Thanks.