To take benefit of the best prices you will need to prove you are not a business traveller, on the way to the airport, so you will need to provide either your flight details or your holiday booking reference. I don't think you will need to give this when you arrive at the station but you may be asked to provide them on your travels from mill-hill-broadway station to the airport.
Going to gatwick airport on the train is great, you can sleep, read (even at night), play travel scrabble, or just stare out of the window... The other caveat to buying these cheap train tickets from the station to gatwick airport is that there must be at least two grown ups travelling together. But another great benefit is that children go free. All children under five go free, and then one child over five but under sixteen per adult travelling by train to gatwick airport.
be sure to leave enough time to get to gatwick airport early enough to eat, shop etc, so check the times of trains from the station before you go. If you end up with spare time at the airport you can always read a book or play travel scrabble again! As you have bought yr train ticket in advance, you don't need to worry about the opening times of the station.
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