Background
A geometry problem, often credited to George Polya, is as follows: the captain of a ship can see three lighthouses and is able to measure the angles from the ship’s position to pairs of lighthouses. The positions of the lighthouses are known. Calculate the position of the ship.
Your task is to locate the “ship”, in this case, a tag that will direct you to the cache.
As a multi
This is a simple tag hunt with 3 tags and the cache. It will take you on a long hike, round trip of 6 to 8 km, through the western part of the South March Highlands.
- At the posted coordinates, there is a film canister that contains the data needed to solve the cache as a puzzle. Use only the location given for the middle lighthouse; ignore the rest.
- At the second tag, you will find the coordinates for the last tag.
- At the third tag, you will find the coordinates for the cache.
- The cache is within 100m of the last tag. There is a faint trail to the area of the cache about 10m west of the tag. The approach is quite steep.
As a puzzle
Although the cache is primarily designed as a multi, you may also find it as a puzzle by using your GPS to find the “ship”. Doing so will shorten the hike to a round trip of about 4km.
Lighthouse A is located at the posted coordinates. There you will find a film canister containing the locations of lighthouses B and C, and the angles, in mils, from the ship to A and B, and to B and C. The ship’s position is unknown; it is referred to as D.
The following is one of several ways to solve this problem. If you use a different method, feel free to describe it in your log. I’d be particularly interested to knowing if anyone is able to use proximity circles effectively on their GPS.
Set your GPS to use mils when solving this problem.
Part 1 uses the central angle property of circles. That states that an angle with its vertex at the center of a circle is twice as large as an angle with its vertex on the circumference, given that both are based on the same arc. Visit this link to see this in action.
In this problem, D lies on the circumference of two circles, ADB and BDC, and is at the intersection of those circumferences. The first step is to find the center of each circle.
- X is the center of the circle containing A, B and D. Because AX=BX, triangle AXB is isosceles, so ∠BAX = ∠ABX = (3200-2θ)/2 = 1600–θ.
- Determine the bearing from A and B, and from B to A.
- From each of A and B, add or subtract (1600-θ) to get the bearing to X.
- From each of A and B, project a waypoint toward X.
- Create a route consisting of A, the waypoint projected from A, B, and the waypoint projected from B. Map the route, and zoom in closely to determine the intersection which is X.
- Repeat to find Y, the center of the circle containing B, C and D. Use (1600-φ) as the angle.
Part 2 uses triangle congruency. Because XB=XD (radii of the same circle), YB=YD (radii of the same circle), and XY is shared, triangles XBY and XDY are congruent; they are reflections of each other. Therefore D is a reflection of B across the line through XY.
- Measure the bearing from X to Y.
- From B, project a waypoint across and perpendicular to XY.
- Z is intersection of XY and BD. To find it, create a route consisting of X, Y, B and the waypoint projected from B. Map the route and zoom in closely to determine Z.
- Measure the distance from B to Z.
- From Z, project a waypoint using that distance and the same bearing used in step #2. The resulting point is D.
Part 3: At D you will find a tag that will direct you to the cache. The cache is located within 100m of D. There is a faint trail to the area of the cache about 10m west of the tag. The approach is quite steep.
Parking and access to the posted coordinates
There is very limited parking, a maximum of 3 cars, at the closest parking coordinates on Huntmar Road. It is not possible to park along the shoulder. This parking is probably not available in winter. You will walk 300m along the road and then follow a short trail down a gravel embankment.
The alternative is to park at the Brady Trailhead on Second Line Road and walk through the highlands. This will add about 5km to your hike.
About the South March Highlands
The South March Highlands are owned by the City of Ottawa and primarily maintained by the Ottawa Mountain Bike Association. It’s a large area with an intricate trail system so be sure to have a map with you, such as this one for your GPS.
Except for the posted coordinates and the final approach to the cache, the route is entirely on trails. The film canister at the posted coordinates is near the end of a branch of a spruce tree, about chest level. The remaining tags are on the backs of tree right beside the trail. The terrain rating reflects the fact that the trails are occasionally rocky and/or steep. The approach to the cache is quite steep. Watch your footing.
The trails are excellent for hiking, mountainbiking, skiing and snowshoeing.