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8
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edited May 14 2011 at 23:57 |
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 $\delta=\pi/8$.
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7
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edited May 14 2011 at 11:37 |
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$ :-)
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6
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edited May 13 2011 at 11:59 |
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 $\delta \ge \pi$ :-)
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5
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edited May 12 2011 at 16:13 |
The analysis
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)?
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4
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edited May 12 2011 at 16:06 |
The analysis
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)?
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3
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edited May 12 2011 at 13:49 |
The analysis
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 draw 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)?
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2
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edited May 12 2011 at 13:29 |
The analysis
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 draw 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)?
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1
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asked May 12 2011 at 13:17 |
Rolling-ball game
The analysis
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 $\delta < \pi/2$.
Player 1 (Blue) rolls the ball to an adjacent
grid point, and the track of the ball-plane contact point
is draw 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)?
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