I really like this problem from the 1985 AJHSME:
Here’s the problem:
In a certain year, January had exactly four Tuesdays and four Saturdays. On what day did January $1$ fall that year?
This problem has always given my son a little trouble. Talking through it tonight he struggles a bit to get going, but slowly gets to the point where he’s drawn a calendar that represents the situation. It doesn’t look like a normal calendar, though.
Around 4:30 in the video he’s talking about why the picture looks wrong, but then comes a really cool idea to fix that problem – re-draw the calendar! From this new point of view the solution to the problem just falls off the page.
What my son went through talking this problem tonight is what learning math often looks like. From one point of view a problem can be nearly intractable and frustrating, but from another everything seems to make perfect sense.