Min of max and max of min

There is no loss of generality in assuming that the numbers are $1, 2, 3, ..., 9.$ It is easy to prove that for a given arrangement of the numbers into the $3\times 3$ - array the condition $\max_j \min_i a_{ij} = \min_i \max_j a_{ij}=b$ is true if and only if the number $b$ is the maximum number of its row and the minimum of it column. The number $b$ is called the saddle point of the corresponding array or matrix. Whenever $b$ is the saddle point there are going to be two other numbers less than it on the same row and two other numbers greater than it on the same column. So the only possible saddle points are 3, 4, 5, 6, 7. Additionally, the way in which the 4 numbers that are not in the same row or column as the number $b$ can be placed does not change the fact that $b$ is going to be the saddle point. Then to form such an array we have to pick a place where the saddle point will be located, this can be done in 9 forms. When the saddle point is 3, the numbers in the same column and in the same row can be selected in $P[2, 2]*P[6, 2]$ ways and the other 4 numbers in 4! When the saddle point is 4, using a similar reasoning, the other 8 numbers can be placed in $P[3, 2]*P[5, 2]*4!$ ways. When the saddle point is 5, the other 8 numbers can be located in $P[4, 2]*P[4, 2]*4!$ ways. When the saddle point is 6, there are going to be $P[5, 2]*P[3, 2]*4!$ ways of placing the other 8 numbers. Finally, when the saddle point is 7, there are going to be $P[6, 2]*P[2, 2]*4!$ ways of placing the other 8 numbers. Therefore the total number of possible arrangements will be $9*4!*(P[2, 2]*P[6, 2]+P[3, 2]*P[5, 2]+P[4, 2]*P[4, 2]+P[5, 2]*P[3, 2]+P[6, 2]*P[2, 2])$ $=108864.$

Let's say that a number is mm iff it is the max-min and the min-max.

Call the number in the center x. x will be mm iff its horizontal neighbours <x and its vertical neighbours>x.

The probability of this is $1/5 \cdot 2/4 \cdot 1/3 =1/30$

We can rearrange the rows/columns without changing the existence of an mm. So the probability of there being an mm is 9*1/30=3/10.

So the answer is (3/10)*9!