Scenario: Arranging boo ks on shelves

You have a collection of $n$ books and $m$ shelves. Each

book must be placed on a shelf, and you want to

organize them in a specific way. However, you can

only arrange books in a straight line either

horizontally or vertically on each shelf.

What is the maximum number of books you can

arrange in the collection, and what is the minimum

number of rearrangements $($clearings$)$ required to

organize all the books? Claim: The maximum number of books you can

arrange is min$($n$,$m$),$ and the minimum number of

rearrangements needed is n$+$m$2.$