http://qs1969.pair.com?node_id=1121627

Before I went to bed, I only had a guess. I started with a pen and paper, and some maths:
w * h = (w - 2) * h + 10 w * h - (w - 2) - h = 10 h * (w - w + 2) = 10 2h = 10 h = 5

Now, we need to find the width. To be able to step on every tile, it must be coprime with the height, and moreover, width - 2 must be coprime with the height as well. I drew the simplest cases and extrapolated the observation to the following sequence: 3, 9, 11, 13, 19, 21, 23...

And when I got up, I verifyied the results with a Perl program. Specify a true command line argument to see the paths:

#!/usr/bin/perl use strict; use warnings; my \$DEBUG = shift; sub steps { my (\$w, \$h) = @_; print "\$w x \$h\n" if \$DEBUG; my \$steps = 1; my @dir = ( 1, 1 ); my (\$x, \$y) = (1, 1); my @visited = ([], [ undef, '\\' ]); while () { \$x += \$dir; \$y += \$dir; \$dir = -1, \$x -= 1 if \$x > \$w; \$dir = -1, \$y -= 1 if \$y > \$h; \$dir = 1, \$x = 1 if \$x < 1; \$dir = 1, \$y = 1 if \$y < 1; ++\$steps; return -1 if \$steps > \$w * \$h or \$visited[\$x][\$y]; \$visited[\$x][\$y] = \$dir != \$dir ? '/' : '\\'; if (\$DEBUG) { for my \$x (1 .. \$w) { for my \$y (1 .. \$h) { print \$visited[\$x][\$y] // ' '; } print "\n"; } sleep 1; print "\n"; } if (\$x == \$w and \$y == \$h) { last if \$steps == \$w * \$h and '\\' eq \$visited[\$x][\$y]; return -1 } } return \$steps } my @steps; my \$width = 1; until (0) { for my \$height (1 .. \$width) { \$steps[\$width][\$height] = steps(\$width, \$height); print "\$width x \$height = \$steps[\$width][\$height]\n" if (\$height > 2 and \$steps[\$width][\$height - 2] = += \$steps[\$width][\$height] - 10) or (\$height < \$width - 1 and \$steps[\$width - 2][\$height] = += \$steps[\$width][\$height] - 10) } ++\$width; }
Update: Spoiler tag removed.
لսႽ† ᥲᥒ⚪⟊Ⴙᘓᖇ Ꮅᘓᖇ⎱ Ⴙᥲ𝇋ƙᘓᖇ