The purpose of Ch'an/Zen meditation is to see our true and ultimate nature, and thereby find perfect freedom and peace.  Buddhists believe that our ignorant attachments to things and viewpoints prevent us from seeing things-as-they-are (which could be called suchness, or emptiness or Buddha nature or Dharma nature or many other names).

Koans are records of how particular masters tried to help students "let go" of certain ignorant attachments, how they tried to awaken them to a certain aspect of reality.  But actually such awakning  is not a matter of finding the one fixed "right way" to look at things.  The point is to be free of all fixed, locked-in, dogmatic perspectives, and to adopt, spontaneously and compassionately, whatever approach helps others to get unstuck from their attachments.  When we begin, we think that good and bad, you and me, high and low, buddha and non-buddha, etc. are really different, so we have to unlearn this.  But if we think that means that these things really the same, then we have still more unlearning to do.  A famous Ch'an saying:  Shakyamuni Buddha is still sitting.

1. The Problem
The great merit of koans is that (according to Zen) they teach the entire dharma in a way that goes directly to the heart.  Each koan has a point, an idea which can be explained.  But in analyzing them theoretically, one misses the point in another sense . . . for one is not grasping directly.  Like a joke:  it can be explained, but then it isn't funny.

2. The Philosophy
In Three Pillars of Zen, Yasutani Roshi is described holding up a paper fan.  One side is printed with a complex pattern, the other side is blank.  The two sides are right there together; neither cancels out the other.  One has complex distinctive forms, one is empty.  All things have two sides like this.  

3. Dharma Nature as one: 
A monk asked Joshu, "Does a dog have Buddha Nature? Joshu said, "Mu!"

The wind was flapping a temple flag.  Two monks argued: One said the wind was moving; the other said the flag was moving.  The Sixth Patriarch said, "It is your mind that is moving."
Commentary:  The Patriarch could not suppress his compassion and made an awkward scene.

/3/Undermining attachment to Oneness:
Master Gutei, whenever, he was questioned, just stuck up one finger.  He had a young attendant, whom a visitor asked, "What Zen does your Master teach?"  The boy stuck up one finger.  Hearing of this, the Master cut off the boy's finger with a knife.  As the boy ran screaming with pain, Gutei called to him.  When the boy turned his head, Gutei stuck up one finger.  The boy was enlightened.

4. Dharma Nature as many
A monk asked for instruction.  Joshu said, "Have you had breakfast?" Monk: "Yes." Joshu: "Then go and wash your bowls."

A monk asked Joshu, "What is the meaning of the Patriarch's coming from the West?" Joshu said, "The oak tree in the front garden."

A monk asked Unmon, "What is Buddha?"  Unmon said, "A shit stick!"

/4/ Undermining attachment to Manyness:
A monk asked Tozan, "What is Buddha?"  Tozan said, "Three pounds of flax."

Commentary:  It is not a lukewarm, "Every single thing is Buddha."  If it were, it would not be praised as a great koan.  Just this three pounds of flax.  Just this live three pounds.  Just this dead three pounds.  Wherever you go it is the same amount!

5. Conventional/Ultimate: Becoming Free to Play the levels
A monk came to be taught, and Isan, seeing him, made as if to rise.  The monk said, "Please don't get up!"  Isan:  "I haven't sat down yet!" Monk: "I haven't bowed yet!" Isan: "You rude creature!"

6.  Practicing Compassion  
It is like a man up a tree who hangs from a high branch by his mouth; his hands cannot reach a bough, his feet cannot touch the trunk.  Another man comes and asks him, "What is the meaning of Bodhidharma's coming from the West?"  How should he answer?

You enter the Founder's room and find a visitor flicking ashes from a cigarette on the statue of Shakyamuni.  What should you do?