It is often hard to judge the difficulty of a problem in discrete mathematics just by reading it. In part this is because it is not always easy to categorize discrete problems, and even when one can, they don't always have routine solution methods.

Therefore we rate problems, from 1 easiest to 5 hardest. The rating appears right after the problem number, in angular brackets. A higher rating can mean more computational difficulty, but it often means the problem involves a significant extension of the reading, or combines ideas from more than one section, or requires a flash of insight. In general, our rating scheme is this:

1. Straightforward. Can be done directly by a method illustrated in this section of the text, and without a lot of work.

2. Middling. Can be done by a method illustrated in this section of the text, but takes more work.

3. Difficult. Typically involves an extension of ideas in the text, or ideas from several sections, or if doable directly from the reading, is quite involved.

4. Very difficult.

5. A real honors challenge. This rating is used very sparingly, mostly in the Supplementary Problems at the end of chapters and in the Final Problems.

When a problem has several parts, only the problem as a whole is rated, and the rating refers to the more difficult parts or the total effort involved for all the parts.

The most common rating is (slightly over half the problems), with the next...