Stealth Ninja

A large palace contains a hall, where the floor is a (2n-1) \times (2n-1) grid of squares, arranged in (2n-1) rows (numbered from $1$ to 2n-1) and (2n-1) columns (numbered from $1$ to 2n-1) — as shown in the figure below for n = 4. Each square that is at the intersection of an even-numbered row and an even-numbered column is covered by a square wooden support column. The other squares are empty. Thus, between the walls and the wooden supports, there are $n$ corridors from left to right and $n$ corridors from back to front through the hall. At the ends of the back-to-front corridors, there are door openings with curtains. Otherwise, the walls are closed.

The hall is guarded by $k$ guards. Each guard is stationed at the intersection of a front-to-back and a left-to-right corridor. Each guard looks for $4$ seconds to the left, $4$ seconds to the back, $4$ seconds to the right, and $4$ seconds to the front, repeating this process indefinitely. The viewing directions of the guards are not necessarily synchronized: while some may look left, others may look forward.

A ninja must traverse the hall from the front to the back without being seen by the guards. Before entering the hall, the ninja waits in row $0$, that is, behind one of the curtains in the door openings in the front wall. The ninja can choose behind which curtain to wait and how long to wait before entering the hall. The ninja can choose any door in the back wall as the exit. To move between two adjacent squares, the ninja needs exactly $2$ seconds — while moving, the ninja can be seen by any guard who is currently watching in that direction. Can the ninja successfully traverse the hall? The guards can see past each other, but not through the curtains.

Figure 1 shows how the ninja finds a way out. First, the ninja waits. After $8$ seconds, the guard in row $5$, column $1$ turns to the left, and the ninja steps out from behind the curtain. After $10$ seconds, the ninja arrives at row $1$, column $1$. After $12$ seconds, the ninja arrives at row $2$, column $1$. The ninja waits until the guard at row $3$, column $5$ turns to the back; then the ninja starts moving and disappears behind the curtain in column $3$ just before the guard in row $1$ turns toward that direction.