MathDB

Let

n> 2

be a positive integer. Given is a horizontal row of

n

cells where each cell is painted blue or red. We say that a block is a sequence of consecutive boxes of the same color. Arepito the crab is initially standing at the leftmost cell. On each turn, he counts the number

m

of cells belonging to the largest block containing the square he is on, and does one of the following:
If the square he is on is blue and there are at least

m

squares to the right of him, Arepito moves

m

squares to the right;
If the square he is in is red and there are at least

m

squares to the left of him, Arepito moves

m

cells to the left;
In any other case, he stays on the same square and does not move any further.
For each

n

, determine the smallest integer

k

for which there is an initial coloring of the row with

k

blue cells, for which Arepito will reach the rightmost cell.

Problem Statement