Second attempt

A group of three SIS students proved themselves at the recent Höjdpunkten Maths Competition. Yebum from Grade 12 and Pablo and Bora from Grade 10 comprised the SIS team that came second in all of Sweden. A total of 91 teams competed, making the SIS achievement a standout performance, and they missed the top spot by just 3 points.

As a flavour of what they contended with here is one of the questions. We would offer to publish they answer but your editor really has no idea... Perhaps a benign mathematician can provide the answer. (Please show your workings...)

Problem 4. Styrbjörn walks around in a grid of size n × m, where n ≥ 2 and m ≥ 2 are integers.
He starts in one of the corners, and every minute he walks into a square next to (i.e. sharing a side
with) the one he is in. Styrbjörn is a bit restless, and therefore he does not want to visit the same
square twice. He also does not want to walk two steps in a row in the same direction. For which
values of n and m can Styrbj ̈orn visit all n · m squares?