Sometimes a good diagram can make all the difference. Consider this problem, 

Duke the dog's head is 6 inches long, his tail is as 
long as his head plus half the length of his body, and 
his body is as long as the head and the tail combined.
which can make your head hurt just reading it. But with the right diagram, 
                                    H = 6
                                   +----+
   T = H + B/2                     |    |
   ------------- +---------------+ +-----
                 |               |
                 |               |
                 +---------------+
                   B = H + T
you're halfway to the solution.

Here's another example where a diagram changes something confusing 

What day of the week was yesterday if 5 days before the day after 
tomorrow was Wednesday?
into something straightforward: 
 +----------+----------+---------+-----+--------+---------+
 |Day before|Day before|Yesterday|Today|Tomorrow|Day after|
 |day before|yesterday |         |     |        |tomorrow | 
 |yesterday |          |         |     |        |         |
 +----------+----------+---------+-----+--------+---------+
      ^
      |
      |________ 5 days before the day after tomorrow

Or consider this problem, 

Every day at noon a ship leaves San Francisco for Tokyo, and at the 
same instant a ship leaves Tokyo for San Francisco. Each trip lasts 
exactly eight days. How many Tokyo ships will each San Francisco ship 
meet?
and this diagram: 

    Day
      |/   /   /   /   /   /   /   /  |
    0 X   /   /   /   /   /   /   /   *
      |\ /   /   /   /   /   /   /   /|
      | X   /   /   /   /   /   /   / |
      |/ \ /   /   /   /   /   /   /  |
    1 *   X   /   /   /   /   /   /   *
      |  / \ /   /   /   /   /   /   /|
      | /   X   /   /   /   /   /   / |
      |/   / \ /   /   /   /   /   /  |
    2 *   /   X   /   /   /   /   /   *
      |  /   / \ /   /   /   /   /   /|
      | /   /   X   /   /   /   /   / |
      |/   /   / \ /   /   /   /   /  |
    3 *   /   /   X   /   /   /   /   *
      |  /   /   / \ /   /   /   /   /|
      | /   /   /   X   /   /   /   / |
      |/   /   /   / \ /   /   /   /  |
    4 *   /   /   /   X   /   /   /   *
      |  /   /   /   / \ /   /   /   /|
      | /   /   /   /   X   /   /   / |
      |/   /   /   /   / \ /   /   /  |
    5 *   /   /   /   /   X   /   /   *
      |  /   /   /   /   / \ /   /   /|
      | /   /   /   /   /   X   /   / |
      |/   /   /   /   /   / \ /   /  |
    6 *   /   /   /   /   /   X   /   *
      |  /   /   /   /   /   / \ /   /|
      | /   /   /   /   /   /   X   / |
      |/   /   /   /   /   /   / \ /  |
    7 *   /   /   /   /   /   /   X   *
      |  /   /   /   /   /   /   / \ /|
      | /   /   /   /   /   /   /   X |
      |/   /   /   /   /   /   /   / \|
    8 +   /   /   /   /   /   /   /   X
      |  /   /   /   /   /   /   /   /|
      SF                            Tokyo
The right diagram can let you replace calculation with counting, or some other simple observation.