8 Another image added, just for intuition.

The analyses in two recent MO questions, "Rolling a random walk on a sphere" and "Maneuvering with limited moves on $S^2$," suggest a Rolling-Ball Game, as follows.

A unit-radius ball sits on a grid point of a $\delta \times \delta$ regular grid in the plane, with $\delta \neq \pi/2$. Player 1 (Blue) rolls the ball to an adjacent grid point, and the track of the ball-plane contact point is drawn on the ball's surface. Player 2 (Red) rolls to an adjacent grid point. The two players alternate until each possible next move would cause the trace-path to touch itself, at which stage the player who last moved wins. In the following example, Red wins, as Blue cannot move without the path self-intersecting.

Q1. What is the shortest possible game, assuming the players cooperate to end it as quickly as possible? For $\delta=\pi/4$, the above example suggests 6, but this min depends on $\delta$. It seems smaller $\delta$ need 8 moves to create a cul-de-sac?

Q2. What is the longest possible game, assuming the players cooperate to extend it as much as possible?

Q3. Is there any reasonable strategy if the players are truly competing (as opposed to cooperating)?

Addendum. We must have $\delta < 2 \pi$ to have even one legal move, and the first player wins immediately with one move for $\pi \le \delta < 2\pi$ :-) (left below).

The right image just shows a non-intersecting path of no particular significance for $\delta=\pi/8$.

The analyses in two recent MO questions, "Rolling a random walk on a sphere" and "Maneuvering with limited moves on $S^2$," suggest a Rolling-Ball Game, as follows.

A unit-radius ball sits on a grid point of a $\delta \times \delta$ regular grid in the plane, with $\delta \neq \pi/2$. Player 1 (Blue) rolls the ball to an adjacent grid point, and the track of the ball-plane contact point is drawn on the ball's surface. Player 2 (Red) rolls to an adjacent grid point. The two players alternate until each possible next move would cause the trace-path to touch itself, at which stage the player who last moved wins. In the following example, Red wins, as Blue cannot move without the path self-intersecting.

Q1. What is the shortest possible game, assuming the players cooperate to end it as quickly as possible? For $\delta=\pi/4$, the above example suggests 6, but this min depends on $\delta$. It seems smaller $\delta$ need 8 moves to create a cul-de-sac?

Q2. What is the longest possible game, assuming the players cooperate to extend it as much as possible?

Q3. Is there any reasonable strategy if the players are truly competing (as opposed to cooperating)?

Addendum. The We must have $\delta < 2 \pi$ to have even one legal move, and the first player wins immediately with one move for $\delta \pi \ge le \pi$ delta < 2\pi$:-) 6 delta >= pi figure The analyses in two recent MO questions, "Rolling a random walk on a sphere" and "Maneuvering with limited moves on$S^2$," suggest a Rolling-Ball Game, as follows. A unit-radius ball sits on a grid point of a$\delta \times \delta$regular grid in the plane, with$\delta \neq \pi/2$. Player 1 (Blue) rolls the ball to an adjacent grid point, and the track of the ball-plane contact point is drawn on the ball's surface. Player 2 (Red) rolls to an adjacent grid point. The two players alternate until each possible next move would cause the trace-path to touch itself, at which stage the player who last moved wins. In the following example, Red wins, as Blue cannot move without the path self-intersecting. Q1. What is the shortest possible game, assuming the players cooperate to end it as quickly as possible? For$\delta=\pi/4$, the above example suggests 6, but this min depends on$\delta$. It seems smaller$\delta$need 8 moves to create a cul-de-sac? Q2. What is the longest possible game, assuming the players cooperate to extend it as much as possible? Q3. Is there any reasonable strategy if the players are truly competing (as opposed to cooperating)? Addendum. The first player wins immediately with one move for$\delta \ge \pi\$ :-)

5 Grammar / typo.
4 Improved figure.
3 Typo
2 Changed to != pi/2
1