Ruby (and Perl) has a concept of the flip flop:
file = File.open("ordinal")
while file.gets
  print if ($_ =~ /third/) .. ($_ =~ /fifth/)
end
which given a list of ordinals, such as
first
second
third
fourth
fifth
sixth
would start printing when it reached "third" and stop when it reached "fifth":
third
fourth
fifth
Is there a functional programming concept analogous to this, or would this normally be described in terms of takewhiles? I'm not asking about a particular language, just what term you'd use to describe it.
In a functional language such as haskell, you would pass in the flip and flop conditions as predicates, and filter an input list based on it. For example, the following is a definition of flipflop in haskell (don't worry about the implementation if you don't know haskell - the key part is how it is used):
flipflop flip flop = 
  uncurry (++) . second (take 1) . break flop . dropWhile (not . flip)
This is how it can be used:
> flipflop (== 3) (== 5) [1..10]
[3,4,5]
It is an example of making an effectively new language construct just by using higher ordered function.
I don't know if there is a special name for that construct in functional languages.
Depends on functional language. How about this?
ff_gen =
  lambda{ |first, *conditions| 
    flipflop = false
    condition = first
    lambda{ |v| 
      if condition && condition[v]
        condition = conditions.shift
        flipflop = !flipflop
        true
      else
        flipflop
      end
    }
  }
ff = ff_gen[lambda{|v| v == 3}, lambda{|v| v == 5}, lambda{|v| v == 7}, lambda{|v| v == 11}]
puts (0..20).select{ |i| ff[i] }.inspect # => [3, 4, 5, 7, 8, 9, 10, 11]
Added: Of course, Ruby is not a pure functional language, so I decided to rewrite it in Erlang:
#!/usr/bin/env escript
flipflop(E, {[H|T] = Conditions, FlipFlop}) ->
  case H(E) of
    true ->
      {true, {T, not FlipFlop}};
    false ->
      {FlipFlop, {Conditions, FlipFlop}}
  end;
flipflop(_, {[], FlipFlop}) ->
  {FlipFlop, {[], FlipFlop}}.
flipflop_init(Conditions) ->
  {[], {Conditions, false}}.
main([]) ->
  {L, _} = 
    lists:foldl(
      fun(E, {L2, FFState}) -> 
        case flipflop(E, FFState) of
          {true, FFState2} ->
            {[E|L2], FFState2};
          {false, FFState2} ->
            {L2, FFState2}
        end
      end,
      flipflop_init([
        fun(E) -> E == 3 end, 
        fun(E) -> E == 5 end, 
        fun(E) -> E == 7 end, 
        fun(E) -> E == 11 end
      ]), 
      lists:seq(0,20)
    ),
  io:format("~p~n", [lists:reverse(L)]),
  ok.
Note: In fact, classic flip-flop should work like dropwhile(!first) -> takewhile(!second), so Ruby's flip-flop is ad hoc one (compare with flip-flop in electronics).
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