Direction Tests

A direction test problem describes a person (or an object) moving in a straight line, then turning through a right angle or more, and asks either how far the person now is from the starting point, or which direction the person is now facing, or both.

For reference, the eight directions in clockwise order starting from North are: North, North-East, East, South-East, South, South-West, West, North-West, and back to North again — each intermediate direction (North-East, South-East, South-West, North-West) sitting exactly midway between the two main directions on either side of it.

The four MAIN directions — North, South, East, West — are arranged, going clockwise starting from North, in the order North, East, South, West; the four INTERMEDIATE directions — North-East, South-East, South-West, North-West — sit exactly between each pair of main directions. Two facts make every direction problem solvable with simple arithmetic:

To find the final position after several straight-line movements, keep a running tally of the NET distance moved in the North-South direction and, separately, the net distance moved in the East-West direction — treating movements to the North and to the East as positive, and movements to the South and to the West as negative. Once both net distances are known, if they are both nonzero, the two form the two shorter sides of a right-angled triangle, and the straight-line ("as the crow flies") distance back to the start is found the same way as any right-angled triangle: if the two net distances happen to form a well-known triple like 3 and 4 (giving a straight-line distance of 5), or 6 and 8 (giving 10), the answer can be read off directly. If one of the two net distances works out to zero, the final position lies exactly along a single direction from the start, and no triangle is needed at all. …