k
complete rows is (k * (k + 1)) / 2
.(k * (k + 1)) / 2 <= n
to find the maximum k
.(k + 0.5)^2 - 0.25 <= 2 * n
.k + 0.5 <= sqrt(2 * n + 0.25)
.k
.k
is an integer, since we cannot have a fraction of a row.k
as the number of complete rows that can be built.