MathDB

In any finite grid of squares, some shaded and some not, for each unshaded square, record the number of shaded squares horizontally or vertically adjacent to it; this grid's score is the sum of all numbers recorded this way. Deyuan shades each square in a blank

n\times n

grid with probability

k

; he notices that the expected value of the score of the resulting grid is equal to

k

, too! Given that

k > 0.9999

, find the minimum possible value of

n

.
Proposed by Andrew Wu

Problem Statement