The dimensions of a box and a book are, respectively, and , as shown in the diagram. What is the maximum number of books that can be packed into the box with no books sticking out?
Solution
We must decide whether the books are to be packed flat, on their spines, or upright (or possibly a combination). In this case it is easy, because if they are packed upright, then their height is , the same as the depth of the box. The width of each book is , which goes twice into the width of the box; the thickness of each book is , which goes six times into the length of the box. Thus upright books will fill the box completely, so no higher number is possible.
Want a route through all this instead of an archive? The track
puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.