Understanding shorthand closure syntax for map fun

Question

I'm trying to understand some of the short hand syntax used by the map function.

The following is the setup

    let array = [1, 2, 3]

    // these make sense
    let arr1 = array.map({String($0)})
    let arr2 = array.map{String($0)}
    let arr3 = array.map({ number in
        return String(number)
    })
    let arr4 = array.map({ (number) -> String in
        String(number)
    })

Here is where the confusion lays. In swift I can forgo the curly braces for map, but this seems like something that can't be done, for my own functions where I have a trailing closure. Some magical inference that's being made perhaps? Also why is the String initialized in this way?

    // this doesn't make sense. Foregoing the curly braces? I can't do that!!!
    let arr5 = array.map(String.init)    
    let arr6 = array.map(String())    // Compile Error: Cannot convert value of type 'String' to expected argument type '@noescape (Int) throws -> _'

This is me trying to use similar syntax as map

func crap(block:(Int)-> String) {
    print("Int to string block" + block(1));
}
// works obviously
crap{ "\($0) some garbage" }
// compile error : Anonymous closure argument not contained in a closure
crap( "\($0) some garbage" )

Answer 1

Distinguish parentheses () from curly braces {}.

In a sense, only the parentheses version is "real", because, after all, that is what a function call requires. In the parentheses when you call map, you put a function. It may be a function reference (i.e. the name of a function):

    let arr = [1,2,3]
    func double(i:Int) -> Int {return i*2}
    let arr2 = arr.map(double)

Or it can be an anonymous function, meaning a function body in curly braces:

    let arr = [1,2,3]
    let arr2 = arr.map({$0*2})

But in that case, and that case only, you can (as a shortcut) use the "trailing closure" syntax instead:

    let arr = [1,2,3]
    let arr2 = arr.map(){$0*2}

But since map takes no other parameters, you can then also omit the parentheses — the only situation in Swift where you can call a function without parentheses:

    let arr = [1,2,3]
    let arr2 = arr.map{$0*2}