$\sqrt{2-\sqrt{2}}$
$2-\sqrt{2}$

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If you draw the path, you get an isoceles triangle with lengths 1. The angle is 45 degrees, since he waddled northwest. We can find the other two angles simply: $180-45=135$ , $\frac {135}{2}=67.5$ . We can solve suing the law of sines: $\frac {sin (45)}{x}=\frac {sin (67.5)}{1}$ . $x=\frac{sin (45)}{sin (67.5)}\approx 0.765$ . $\sqrt {2-\sqrt {2}}\approx 0.765$ .