The Bears
The standard number of bears is three. At the beginning of the game, after the players have placed all of their men and before they start moving them, the bears are moved three times (for an additional dose of unpredictability).
After that, they always move at the end of each round after the last player has completed his turn. Conveniently, this player should also be the "bear-master", i.e. the one to roll the dice for the bears and execute their moves on the board. Since this can be a bit complicated for beginners, for the sake of speed the last player should also be the most experienced one.

Bear Movement
While the players'men move only on the trail marked by the circles, the bears are free to roam the entire board (with the exception of certain obstacles). They move on the squares, in eight possible directions (horizontal, vertical, and diagonal).
At the beginning of the game, the order of movement of the three bears is agreed upon. (This can be important when they get in each other's way.)
The bear-master rolls a 4-sided and a 6-sided die. The resulting roll is valid for all three bears. The number on the 4-sided die determines the direction in which the bears are heading, and the 6-sided one the length of their progress in that direction.

The latter is extremely simple: a "1" on the 6-sided die means a move of one square, a "2" a move of two squares, etc.

As for the direction, the key is as follows:

                1:            45º to the left
                2 or 3:    straight ahead
                4:            45º to the right

For bears with a visible preference for one side over the other (expressed by the - permanent! - orientation of their heads), these rules have to be modified slightly.

A bear that looks to the left moves as follows:

                1:            90º to the left of its body axis
                2 or 3:    45º to the left of its body axis
                4:            straight ahead (in the direction of its body axis)

A bear that looks to the right moves like this:

                1:            straight ahead (in the direction of its body axis)
                2 or 3:    45º to the right of its body axis
                4:            90º to the right of its body axis

Before a bear starts trundling along, common sense dictates that the longitudinal axis of its body be orientated in the corresponding direction:

Complicated? A little, at the beginning... But bear with me [Gasp! ed.], there's more to come:
What happens when an obstacle (mountain, water, station, another bear, or the border of the board) prevents a bear from moving the whole length of its move in the assigned direction?
Answer: No movement points are lost. The bear will just have to change direction according to a neat set of additional rules, the underlying principle of which is the animal's overbearing [Stop it this very minute! ed.]  reluctance to waste precious energy:
1. If there is ONE alternative direction which deviates less degrees than the others from the present one, the bear will unhesitatingly plunge for it:

2. In the case of a draw between two directions for least deviation, the bear will choose the direction which offers the longer stretch of unhindered progress:

3. Should there still be a tie, the square situated closer to the centre of the board, and failing that, the square further away from the border of the board takes precedence.

In the above list of obstacles water was included, which is basically correct, but there's one important exception (and there had better be, after all bears are excellent swimmers!).
A bear will enter into the water, if both of the following conditions are fulfilled:

1. Before its move, the bear must stand immediately adjacent to the water.
(This is why the polar bear in the example we saw earlier did not enter the water.)

2. The 4-sided die (which determines direction) has to point the bear directly to the water. The word "directly" is important here; it implies that in the case of a bear being stopped by "other" obstacles, water squares are not taken into consideration as "alternative directions" in the sense of the above set of rules:

The water slows the bears down drastically: All moves into / within / out of the water automatically have a length of "1", in other words, the number on the 6-sided die is ignored in these cases.

For beginners, understanding and correctly applying bear movement is easily the most challenging aspect of the game, but with patience and a little practice under the guidance of an experienced bear-master you, too, should be able to ensure that the furry brutes won't lose their bearings [That's it, you're fired! ed.].

Man Eaters
At any time, a bear can "see" (or "hear", or "smell"...) only the three squares directly in front of it, in other words, the ones it could enter next:

If there is a man on one of them, the bear forgets about its original direction, enters the square in question and gobbles up that poor loser:

Should there be more than one man on the square, they are all eaten up, no problem.
If there are men on more than one of the three squares in question, the same principles apply as for movement in general: The bear won't change direction unless it has to, and then opt for the direction which least deviates from the original one (with the same additional rules for cases of drawn directions as above).
Obviously, the unfortunate victims can take no further part in the action and are permanently removed from the game. The bear stops to rest and digest on the square of the kill, forfeiting remaining movement points.