Why does Milinda step into Nagasena's trap? Referring to the discussion between Milinda and Nagasena: 

Why King Milinda does not consider all together - pole, axle, wheels etc. - his chariot? Normally one defines a chariot a means for transportation consisting of pole, axle, wheels, etc. This would be the correct answer to Nagasena's question what the chariot is.

Instead, when pressed by Nagasena he names chariot only a designation, a description, an appellation, a name. That is, Milinda rejects the correct answer. Now Nagasena has an easy time to defend the Buddhist principle of anatman.
If I remember correctly, the traditional interpretation of this is that a pile of components on the ground is not the chariot either.

I suppose in modern language we could say that "Chariot" is a designation for the function that depends on the right configuration AND on the external supporting factors like the road, the horses, the gravity, the driver, the passengers etc. Plus, if individual components of the chariot are replaced with functionally equivalent but non-identical ones - the new chariot is neither the same nor different. Furthermore, the wheels etc. are only designations for things that can perform function of a wheel etc. - not to those particular wheels.

So the functional nature of the chariot arises from a number of functional conditions coming together - for which "the chariot" is only a label.

A similar illustration can be found in Vajira Sutta:


  So you believe there's "a being"?
  -- this is a view of Mara.
  In a pile of mere components (sankhara)
  no any such "being"'s found.  
  
  Just as they say 'chariot'
  of an assemblage of parts,
  So they'll call "a being"
  the mere heap of components (skandha).  


The key word here is "an assemblage" or in another translation "when the parts are rightly set". So that's why they are saying that the chariot is not all its parts together either - because the parts by themselves do not produce a function (of a chariot, a being etc), the right combination (of conditions) does.

But on the other hand, there is no "secret sauce" either - nothing apart from the factors and components that would set things in motion: 

(from Viticcha Jataka)


  The Ascetic tried to catch him in his words: "What is the Ganges? Is the sand the Ganges? Is the water the Ganges? Is the hither bank the Ganges? Is the further bank the Ganges?"
  
  But the future Buddha said to him: "If you take exception to the water, the sand, the hither bank, the further bank, where can you find any Ganges River?"


The "neither found in any individual part, NOR in the pile of parts, NOR apart from the parts" thesis is probably constructed after canonical Buddha's line in which he shows how such phenomenon as the "Tathagata" can neither be found in the five skandhas ("piles") nor apart from them:

(From SA 105)


  [The Buddha] asked again: "How is it, Seniya, is bodily form the Tathāgata?"  
  
  [Seniya] replied: "No, Blessed One."
  [The Buddha asked again]: "Is feeling … perception … formations …
  consciousness the Tathāgata?"
  [Seniya] replied: "No, Blessed One."
  [The Buddha] asked again: "Seniya, is the Tathāgata distinct from
  bodily form? [32b] Is the Tathāgata distinct from feeling … perception
  … formations … consciousness?"
  [Seniya] replied: "No, Blessed One."
  [The Buddha] asked again: "Seniya, is the Tathāgata in bodily form?
  Is the Tathāgata in feeling … perception … formations … consciousness?"  
  
  [Seniya] replied: "No, Blessed One."
  [The Buddha] asked again: "Seniya, is bodily form in the Tathāgata?
  Is feeling … perception … formations … consciousness in the Tathāgata?"  
  
  [Seniya] replied: "No, Blessed One."
  [The Buddha] asked again: "Seniya, is the Tathāgata without bodily
  form … feeling … perception … formations … consciousness?"
  [Seniya] replied: "No, Blessed One."  


In my understanding, all of this is a finger trying to point out the fact that reality underlying conventional expression can never be exhaustively described, that our attempts to represent the reality is always an exercise in isolating individual factors and reassembling them into a model, like projecting a multidimensional figure on a two-dimensional surface - with a label (nama) stamped on top.

From taking our models seriously ("reifying" them) comes trying to make them our base, leaning on them, identifying with them, attaching to them - but because our models are merely designations for the fluctuating soup of factors - to assume them to be stable is an inherently faulty assumption - and from this comes dukkha. When we see reality as reality and models as models, we can stop confusing the two, stop getting attached to models, stop trying to establish our base in a model - so from this right vision (of the ineffable reality and the modeling activity) comes baselessness and from this baselessness comes cessation of nutrition of old dukkha and non-production of new dukkha.