java: Arrays.sort() with lambda expression

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The cleanest way would be:

Arrays.sort(months, Comparator.comparingInt(String::length));

or, with a static import:

Arrays.sort(months, comparingInt(String::length));

However, this would work too but is more verbose:

Arrays.sort(months,
            (String a, String b) -> a.length() - b.length());

Or shorter:

Arrays.sort(months, (a, b) -> a.length() - b.length());

Finally your last one:

Arrays.sort(months, 
    (String a, String b) -> { return Integer.signum(a.length() - b.length()) }; 
);

has the ; misplaced – it should be:

Arrays.sort(months, 
    (String a, String b) -> { return Integer.signum(a.length() - b.length()); }
);

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