Imagine there's a permanent cache at N 89° 40.897' W 142°18.120' which is just 21.965 miles from the north pole. To get there from the posted coordinates we could head mostly north through our UTM zone 15, i.e. through bands T,U,V,W and X. At 84° N the UTM system stops and is replaced by its companion system known as UPS (Universal Projection Stereographic). In UPS we arrive in zone Y which covers the entire western half of the polar cap. Just as our zone 15T is subdivided into 100km by 100km squares so is zone Y. Continuing toward our destination we would pass through sub-zones SH, TH, UH, XH, and YH before getting to ZH where the cache is. One corner of ZH is the north pole. Our make-believe cache is located at Y ZHeeeeennnnn whereas our real cache is at 15T VKeeeeennnnn. This is really all you need to know to be able to solve the puzzle; however the following may prove useful.
Figure 1 shows a projection of the northern hemisphere. It's oriented with the prime meridian at the bottom, the international date line at the top, 90°W is to the left and 90°E to the right. The 60 UTM zones are shown in addition to the polar region. The dotted circle is the 45th parallel and the dotted radial line is the 93rd meridian; they intersect in our UTM zone 15T.
As with UTM a rectilinear coordinate system is desired. To get one a secant plane is positioned below the north pole. Points from our earth model are projected onto the plane. The earth model is the same ellipsoidal model used by UTM and the usual lat/long . The projection is from the south pole, through the ellipsoid onto the plane. A secant plane is used rather than a tangent plane to minimize distortion.
Figures 2 and 3 have the same orientation as Figure 1, which is standard in UPS and grid systems based on it, such as MGRS (the Military Grid Reference System) . Click on the icons for these figures in the gallery to the right above or in the list to the left below.
Figure 2 shows the 100km by 100km squares covering the northern polar region. We're interested in those coming in from the left, just above 90° W, viz. zones SH, TH, UH, XH, YH and ZH .
Figure 3 shows zone ZH and the stereographic projection of a our point which is labeled C. Note that zone YZH is rotated by 90° compared to our zone15TVK. The pole is in the lower right hand corner at P. The grid lines within ZH are every 10km in each direction. Thus the lower left corner is (0,0) and the upper right corner at (100,100) in kilometers. The horizontal direction is still referred to as "easting" although its actually north at the bottom of ZH and west at the top, and the vertical direction is referred to as "northing" although its actually west on the left edge of ZH and south on the right edge. The coordinates of C are labelled by X and Y as shown.
To check your solution, enter it in the coordinate checker in the form VKeeee3nnnn0. Note that the final digits of both coordinates, viz. 3 and 0, are provided as some slight rounding error may result in your computations. The correct solution will return the corresponding lat/lon form as well as a hint on the hide.
You can validate your puzzle solution with certitude.
The first 10 puzzle solvers are:
1. rickrich
2. ctc128
3. NYWanderer
4. sparkyfry
5. salsman
6. mnboyscout
7. Briarwoodhill Clan
8. foundinthewild
9. rustynails.
10. Bunganator