The largest sum of the absolute value of difference between each adjacent pairs, of an arrangement in a row of $1$ to $n$.

Question

The initial question is from BMO R1 2012, which is easy.

Consider the numbers 1, 2, . . . , n. Find, in terms of n, the largest integer t such that these numbers can be arranged in a row so that all consecutive terms differ by at least t.

The answer is obviously the floor of n/2.

But if I change the question, say,

Consider the numbers 1, 2, . . . , n. These numbers are arranged in a row Find, in terms of n, the largest number m such that m is the sum of the absolute value of differences between each adjacent pairs.