The four stages involved in the intuition process :
Preparation : we skip on the question
Incubation : we leave the problem and engage in other activities while our unconscious takes over
Illumination : the unconscious finds a promising path and suddenly communicates it to the conscious
Verification : we explore (through awareness) the intuitive path we have just obtained

Question resolution
According to Hadamard, the job of the unconscious is to make combinations of ideas in open way than the conscious does.
If the conscious only searches for ideas related to the problem being addressed, the unconscious can work with ideas from elsewhere, without direct connection to the subject.

Invent is to choose
The subconscious explores a wide range of possibilities. An illumination occurs when it comes across a "remarkable" combination and communicates it to the conscious. The appearance of an intuition is therefore ultimately the result of a selection : a set of "remarkable" ideas combinations.

The criterion by which the unconscious judges the remarkable character of a combination of ideas is the elegance : the mathematical beauty of the combination.

Aspects involved in a mathematical result perceived as elegant
- Symmetry (invariant) : an object (abstract or geometric) which is invariant under certain transformations translates the idea that this object contains something fundamental and essential.
- Generality of a result : generality translates an idea of ​​"power" which contributes to the elegance of the theorem.
- Unity of a result with the theory that encompasses it. Certain results link mathematically a priori very distant entities. The fact that these entities are found in the same result is surprising and spark admiration.

Example : not remembering the name of a person you know
It happens during a discussion that you wanted to mention the name of a person but it does not come into your mind.
Despite the various attempts (sometimes you're very close, and it's almost on your lips) you can't seem to find it.
Then you stop looking for it, and during the discussion, it suddenly comes to your mind. It was always stored somewhere in memory, a back ground process found it for you.