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On Formal Nights they would be expected to wear a jacket and tie - take their school blazers or buy them a blazer. The American children look very smart - some even have their own tuxedos.
Probably only a max of 3 formal nights and if they take part in the excellent kids clubs (my girls loved them at 14 & 15 previously)they often eat elsewhere whilst you relax at dinner. If they do go you really can get away with putting kids that age in a nice shirt (tie optional) and trousers and shoes. They can change straight after dinner. Yes jackets are the dress code and I personally get annoyed at ADULTS who don't at least wear a suit and tie but to get boys out of training shoes is an achievement. We're doing the Explorer early 2009 can't wait - we love Royal Caribbean.