Puzzle #91: The Library

Great Uncle Henry died and left you his books and a container for them (called a mansion). In one of the mansion’s many comfy libraries are books on many subjects.

There are 54 books on birds, 75 on abucuses, 15 on automobile repair, 19 on linguistics, 14 on linguini, 28 on etiquette, 89 on drawing, 32 on fire, 108 on Scotland, 65 on French dialects, 25 on law, 38 on swimming, 12 on diving, 32 on agriculture, 13 on triskaidekaphobia, 42 on rocketry, 3 on wishes, 5 on teddy bears, 17 on U.S. customs, 12 on U.S. Customs, 6 on lakes, 15 on rivers, 78 on computer science, 42 on organic chemistry, 77 on inorganic chemistry, 15 on windup toys, 15 on games, 6 on planets, 63 on vegetarianism, 65 on cats, 32 on dollies (toys), 16 on dollies (handcarts), 4 on libraries, 31 on history, 12 on plush toys, 19 on rolltop desks, 58 on German dialects, 42 on tourism, 75 on business, 23 on sailboats, 9 on credit cards, 14 on fireplaces, 34 on fire suppression systems, 23 on vampires, 17 on werewolves, 29 on castles, 43 on woodworking, 17 on lawns, 22 on duplication, 7 on number theory, 14 on abstract algebra, and 42 cheap, bodice-ripper romances.

Each book has at least one subject. Assume that each subject is independent of the others. For example, a book on linguistics is not necessarily a book on French dialects and vice versa. If a book can have no more than three subjects, what is the minimum number of books in this library?

Solving the above would involve a lot of work. You possibly do not have time to run through the millions of millions of possibilities, so the puzzle is actually: How would you go about solving this problem reasonably efficiently?

Submit your answer to Gene Wirchenko <genew@telus.net>. Your answer should be in the form of a proof. That means to show how your answer must be correct. The deadline is Wednesday, March 4, 2015 at noon Pacific Time. I will post the answer shortly after.