Drawing Arrows - Math Problems | Repovive

Drawing Arrows Beta

There are $n \ge 1$ dots on a sheet of paper, initially with no arrows.

Alice and Bob take turns drawing an arrow from one dot to another distinct dot. Alice goes first. They must follow two rules:

Each pair of dots may be connected by at most one arrow, regardless of its direction.
No dot may have both an incoming arrow and an outgoing arrow.

The player who cannot draw a legal arrow loses.

Determine who has a winning strategy for each positive integer $n$ .

Write proof

You can write formulas in LaTeX or in plain text, and write your solution in any language.

There are $n \ge 1$ dots on a sheet of paper, initially with no arrows.

Alice and Bob take turns drawing an arrow from one dot to another distinct dot. Alice goes first. They must follow two rules:

Each pair of dots may be connected by at most one arrow, regardless of its direction.
No dot may have both an incoming arrow and an outgoing arrow.

The player who cannot draw a legal arrow loses.

Determine who has a winning strategy for each positive integer $n$ .

You can write formulas in LaTeX or in plain text, and write your solution in any language.