Direction and distance questions come every year in OPSC ASO, usually 2 to 4 per paper. They come in four forms: walking paths, turning by angles, shadows and renamed directions. One method solves all of them: draw it.
The compass
North is up, East is right, South is down, West is left. The in-between directions are North-East, South-East, South-West and North-West. Neighbouring directions in this list of eight are 45° apart.
- Right turn = 90° clockwise: N → E → S → W → N.
- Left turn = 90° anticlockwise: N → W → S → E → N.
1. Walking paths: use coordinates
Start at (0, 0). Going East adds to x, West subtracts from x, North adds to y, South subtracts from y. At the end:
- Distance = √(x² + y²). The common right triangles are 3-4-5, 6-8-10, 5-12-13, 8-15-17 and 9-12-15.
- Direction comes from the signs: (+, +) North-East, (+, −) South-East, (−, +) North-West, (−, −) South-West. If one coordinate is 0, it is a pure direction.
ASO 2021: 20 N, right 30, right 35, left 15, left 15 → x = 45, y = 0 → 45 m East.
Trap: "from the back side of a house whose door faces East" means he starts walking West.
2. Turning by angles
Add the clockwise turns and subtract the anticlockwise turns, then turn once by the net amount.
ASO 2021 and 2025 (asked twice): facing West, 45° clockwise, then 180° clockwise, then 270° anticlockwise. Net = 45 + 180 − 270 = −45°, so he ends facing South-West.
3. Shadows
- Morning: the sun is in the East, so shadows fall to the West.
- Evening: the sun is in the West, so shadows fall to the East.
- At noon there is almost no shadow, so OPSC does not ask it.
ASO 2018: in the morning a shadow falls to his right. West is on your right only when you face South.
4. Renamed directions
"If South-East becomes North …" means every direction is turned by the same angle. SE → N is 135° anticlockwise, so West → South-East (ASO 2015 and 2018).
5. Relative positions
For a statement such as "A is West of B, C is East of A but West of B", put the names on a line from West to East (ASO 2015).