Common Sense Evaluation

First, the library can be simplified into just two books. If there are two books, the greatest number of words possible is one and the other must have zero.

2 books: 1 word and 0 words.

Since the number of books is greater than the number of words in the largest book without repetition in number of words, the number of books will always be one more than the number of words in the largest book. There will always by necessity be books with every number of words possible less than the largest book, and there will always be a book with no words to increase the total number of books. This assumes (correctly) you can’t have books with a negative number of words.

Following this logic, a library with three books has a book with two words, a book with one word, and a book with zero words.

3 books: 2 words, 1 word, and 0 words.

A library with n books has a book with n-1 words, a book with n-2 words, a book with n-3 words, and eventually a book with n-n words or zero words.

n books: n-1 words, n-2 words … 0 words.

via