When taking over some new classes this semester, my new students were quite upset when I told them how I managed homework. Their previous teacher assigned only even problems from their book. In most books today, the evens are the students favorite because they have answers in the back of the book so they can "check" their answers. Or, if they can't quite figure out the answer, they can look and try to reverse engineer it.
I don't particularly mind this. Checking the answer lets you know if you made a mistake and, if you did, you get practice finding mistakes as well as correcting them. Reverse engineering problems makes flexible thinkers who can view math as a set of tools that goes two ways instead of a strict linear process that can only be applied one way in one situation.
Still, I don't like students to rely on this too much. They argue that homework is still learning (which it is) so they should have as much help on it as humanly possible. I remind them that they already have a pretty phenomenal amount of resources available to them: They have my lecture (from which I hope they took notes), the book complete with examples that are exactly the same with swapped out numbers, and in many cases, each other. So I don't feel too much pressure to give them all even numbers.
So on this first homework assignment I gave a mix of odds and evens. Some they could "check". Some they couldn't.
In general, I found two common patterns for mistakes. The first was simply not reading the instructions. When it asks how many solutions a set of linear equations has, that question needs to be answered. Don't just draw the graph and leave me to do the rest of your thinking by seeing it only has one, or none, or an infinite number.
The second common mistake was (oddly enough) only on even problems. It was students having the answer to the next even problem. They'd written the problem right, gone through some algebra voodoo magic, and then amazingly arrived at the answer for the next problem. The only way I can explain this is that they did what they felt was enough work to arrive at the answer, "checked" it, wrote down the book's answer but because the answers are all squished together in the back, wrote down the wrong line.
So much for actually "checking" anything or taking that solution and reverse engineering it to meet your work. I'll be sticking to the evens now.