Bears Can Swim, Can't They?

Hint: A bear travelling straight without turning or curving right or left is traveling along a great circle, not to be confused with a typical latitude (except the equator, which is a great circle).

$\textbf{See graphic of Northern Hemisphere, with equator and North Pole. Spherical}$ $\textbf{triangle PQR shown here has vertex angles ∠P = 20°, which is arbitrary,}$ $\textbf{∠Q = 90°, ∠R = 90°, and has great circle arcs PQ = 90°, QR = 20°, RP = 90°}$ $\textbf{Great circle arcs PQ and RP must necessarily extend 90° on the globe.}$ $\textbf{The bear can start at P anywhere in the Northern Hemisphere, then travel}$ $\textbf{towards the North Pole and then over it, continuing southward until he has}$ $\textbf{traveled 90° on the great circle before turning left at Q, thence to R, turning}$ $\textbf{left again, and returning to P. Since P can be anywhere in the Northern }$ $\textbf{Hemisphere, and if the bear's color depends on its starting location P, then}$ $\textbf{its color cannot be determined.}$