JBMO 2026 Problem 3

Let $n \geq 3$ be an integer. There are $n$ colored lamps arranged in a circle. Pressing the button of a lamp once changes its color as follows:

• green $\rightarrow$ red,
• red $\rightarrow$ blue,
• blue $\rightarrow$ green.

Initially, all lamps are colored red. Aladdin makes moves on these lamps. Each move consists of the following three steps:

• he chooses a lamp L, without pressing its button;
• he presses the button of the clockwise neighbor of L once;
• he presses the button of the counterclockwise neighbor of L twice.

For each $n$, determine the maximum possible number of lamps that are colored green simultaneously, after finitely many moves.