Crosswords are a great example of problem solving. If you want to build a framework for understanding problems, crosswords are a very good place to start.
Crosswords are problems we solve for fun. We may learn a little bit from them, but most of the benefit we derive comes from the experience of solving them. It feels good to solve a problem. People don’t usually have fun solving a problem which is too simple (boring) or to difficult (frustrating). It’s fun to tackle problems that push, but don’t exceed (if we’re willing to invest a little thought/effort) the scope of our ability. We generally call this range of difficulty “challenging.”
Crosswords are, generally speaking, fairly well specified problems. A crossword is heavily, but not completely constrained. The author has a specific solution in mind, and gives enough clues to describe that solution well.
A fun game
If you want to pose yourself a real challenge, try and solve a crossword multiple ways. Solve it, then try and find a different solution which still satisfies all the clues. Is there any portion of the grid for which the clues could possibly be satisfied by a different combination of words?
Probably you won’t be able to find another solution, but it really depends on the puzzle in question. Crosswords are written by people, and people aren’t perfect. If some of the clues are vague or ambiguous, you may be able to come up with a valid solution that they never imagined. English is an enormous language with many esoteric words that they might not have been thinking of 
Particularly, if the crossword is loosely connected, you may be able to find a couple words you can swap out.
How else could you describe a crossword?
Most problems can be described in many different ways. You can describe any crossword without a grid, if you want to, by describing the constraints enforced by the grid in a different way. E.g.
Word 1:
Anger
4 letters
Letter 2 is the same as letter 4 in Word 7
Word 2
comfort or encouragement
6 letters
Letter 3 is the same as letter 1 in Word 4
Letter 6 is the same as letter 2 in Word 3
etc, etc, etc.
This description is more flexible than a grid. Any grid crossword could be described like this, but not every crossword that you could describe like this will fit in a grid. For example, the crossword with:
Word 1
Farewell!
3 letters
Letter 1 is the same as letter 2 in Word 4
Letter 2 is the same as letter 4 in Word 7
Letter 3 is the same as letter 4 in Word 6
Letter 3 is the same as letter 1 in Word 2
Letter 3 is constrained twice, which is not impossible (but it does mean that letter 4 in 6 and 1 in 2 must be the same letter, in this case, “e” for “Bye”) The puzzle is not logically inconsistent, but there’s no way to draw that configuration on a 2D grid. It could be drawn on a three way grid.
You could describe a crossword by focusing on the letters, giving each populated (letter) square a number:
E.g. a 10×10
Clue one: letters 1,2,,3,4
Clue two: letters 2,12,22,32
Again, this is more flexible than the grid.
If Software Engineering was a Crossword Puzzle
Another fun form of problem solving is software development and design. Let’s see how it compares with crosswords by imagining we solved crosswords the same way we design software packages shall we?
The newspaper would only have a portion of the crossword grid graphically displayed. Different clues of the crossword would be printed on different pages of the newspaper and you’d have to piece the crossword description together yourself.
Clues would look like:
4 Down: A few letters long. Should rhyme with orange. Postpone until Version 2.
7 Across: A synonym for hippopotamus.
10 Across: Talk to Bob in Accounting.
9 Down: This word should fill the 9 down slot.
Rather than the newspaper posting a solution, you would submit yours to them for approval/disapproval. You would include some notes, like,
“You said that 4 down should be postponed until version 2, but 8 across depends on 4 down, and you’ve marked it high priority.”
“Elephants and hippopotamuses are genetically very very similar.”
Sometimes you would add words to the puzzle that you thought were nifty words, on the off chance that the newspaper would adopt them as solutions.
The newspaper would respond to your solution and tell you which answers were incorrect. You would then resolve the puzzle based on their feedback. This back and forth would continue until you came up with something that they liked, or until they were too tired to argue about it.
The first three paragraphs of this post are a fine example of very crummy writing.