Kōan
A koan (from Japanese 公案 Kōan) is a paradoxical statement or question used mainly in Japanese Zen Buddhism as a meditation technique for novices, especially of the Rinzai school. Koans, Zen, but here we will quote from all spiritual traditions, are as enigmatic as they are fascinating and boast a thousand-year history. We will delve into their origins and characteristics here, and this section will serve as a container for all the most famous koans and others that the Founder will deem fit to submit to your attention.
In Japanese the term literally means "public case" or "announcement". Like almost all Japanese terms it is derived from the Chinese kung-an and originally referred to an edict (notice) issued by a government office of the Chinese empire. By extension, it is a short phrase, question or answer of a paradoxical or enigmatic nature and may refer to stories, parables, short statements or even a few words in a sentence that refer to a story used in practice by Zen masters. These aphorisms may be taken from sayings or stories of ancient or modern Buddhist [and other] masters. A Koan can be studied both from a historical perspective and as a contemplative tool for meditative practice.
In the training of Zen monks, it is a fundamental practice of meditation, as it challenges the student to reflect beyond his usual (usually unconscious and preset) modes of thinking and reasoning.
Traditionally, the teacher gives the koan to the student, who must reflect on the statement or question. The answer to the koan is the student's object of meditation, which will engage him in his daily practice. A koan cannot be understood or answered in conventional terms: it requires a student to abandon dependence on his ordinary modes of understanding in order to access the path to enlightenment.
A characteristic example is this famous koan:
“When you clap both hands you make one sound; what is the sound of just one hand clapping?”
Sometimes the koan is set in the form of a question and answer, for example:
“Is there a sentence that is neither right nor wrong?” “A piece of white cloud shows no flaws.”
Based on what has been said so far, one wonders why this practice is called a “public case”. What makes a koan “public” is the invitation to everyone to observe reality in the here and now. Reality is in fact democratically and immediately available in the present, without the need for a master to transmit, indicate or teach it. So nothing is more public than reality itself. But since there is the infamous Plato's cave, which still today forces us not to see reality but a distorted reflection of it, koans are simple tools to reveal this truth and help the mind to free itself from its habitual thought patterns to arrive at a greater understanding of things, approaching enlightenment.
In current Western "culture" they are often defined as riddles, a term that not only distorts them, but also highlights our tendency to always seek solutions through conceptualization and the ordinary mind. A koan is not a problem to be solved, nor does it require arriving at an answer through the mind. On the contrary, it allows us to understand that that same mind and conventional concepts can never provide us with a satisfactory answer.
The exchange between masters and apprentices is much deeper than the common problem-solution mechanism that the school system tends to teach. It is no coincidence that many Western scholars struggle to understand the meaning of koans, labeling them as irrational and claiming that they row against the intellect and glorify impulsiveness. Instead, koans actually aim to go beyond reason, but not to destroy or deny the intellect. They simply help us understand that reality cannot be captured and condensed into a thought or a precise definition. Seeing reality as it is, and not as the mind wants to shape it, is the true invitation of the koan. The intellect aims to build models of reality, but can never represent it in its entirety. Through the paradoxical and inscrutable nature of the koan, we can take off our blinkers and simply observe the world without chasing prefabricated answers and definitions that distance us from enlightenment.
The point is to broaden the mind by going out of the usual habits trying to bypass intellectual or realistic thinking. Once you are able to look at a Koan not as a question to be answered rationally, you can approach it and find your own answer.
Many see that even in the teachings of people like Jesus Christ, or Mohammed or other illustrious masters one can find innumerable koans.
Like the practice of Zen, some in Christianity believe that one cannot simply “interpret” the words of Christ intellectually.
Instead, we need to go deeper and reflect on some of his statements in order to arrive at a deep spiritual understanding of their meaning.

Bibliography:
The Ditto and the Moon – Alejandro JodorowskyThe Illumination in Daily Life – Engaku Taino Management by Zen Koan – Tetsugen Serra (Zen Master)