Mr. Riek starts from Point A and walks 22 km towards North. He then takes a left turn, walks 30 km, turns left and walks 28 km. He then takes a left turn and walks 39 km. He takes a final left turn, walks 6 km and stops at Point P. How far (shortest distance) and towards which direction should he walk in order to reach Point A again? (All turns are 90 degrees turns only unless specified)
9 km to the west
Place A at the origin (0, 0), with east as +x and north as +y. Walking 22 km north takes him to (0, 22).
A left turn while facing north means facing west; 30 km west takes him to (–30, 22). A left turn while facing west means facing south; 28 km south takes him to (–30, –6).
A left turn while facing south means facing east; 39 km east takes him to (9, –6). A final left turn while facing east means facing north; 6 km north brings him to P = (9, 0).
| Leg | Direction | Distance | Position after leg |
| 1 | North | 22 km | (0, 22) |
| 2 | West | 30 km | (–30, 22) |
| 3 | South | 28 km | (–30, –6) |
| 4 | East | 39 km | (9, –6) |
| 5 | North | 6 km | (9, 0) |
P lies at (9, 0), i.e. 9 km due east of A. To return to A he must walk 9 km towards the west.
Check: net east–west movement = 39 – 30 = 9 km east; net north–south movement = 22 + 6 – 28 = 0. So P is exactly 9 km east of A and the way back is 9 km west.
Hence, 9 km to the west.
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