I don't know whether it's true or not (I can't image that it would, unless it was a monster update or something) but if you bought the game during Alpha (like I did) then you won't have to pay for any future updates. That was the promise Notch gave at the time.
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I've tried it in Chrome, Firefox and Opera now and none of them work for me. Sounds like you just go lucky :/
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I'm getting the same issue.
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