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Ask for a dining table with at least 8 seats (the bigger the better) as this will allow you to meet up with quite a few people and also avoid being permanently stuck with someone who you just don't get on with. Having set times is a good option as you can develop a rapport with the same people as the days progress. You don't say which cruise line you are going with but I've travelled by myself with P&O on a number of occasions and our table group often went to the evening quiz night as a single team, or to a show followed by a drink in the bar. You also get plenty of tips on where to visit in the forthcoming ports and if your table companions are on the same tour they often ask if you would like to meet up and be on the same coach. Don't be concerned if you are on a table full of couples. They have seen each other for months on end and are glad to speak to someone else for a change! Have a good holiday (and may I be the first to wish you a Happy Christmas).
A great way to see the Caribbean Islands is by Taxi picked up on the quayside. These usually take the form of a people carrier. Try to get together with a few people from your dinner table, or anyone else you get friendly with, and hire a taxi on each island to see the sites or go to the beach, etc. Get a book, or listen to the port talk, and then make your mind up where you would like to go and see. Tell the taxi driver and agree a price. They will take about six of you around to see the sights at a fraction of the price of a ships tour! You only pay them when you get back to the ship, so they will not leave you stranded or anything. Make sure you plan to get back at least an hour before the ship leaves as it will not wait for you if you are late! Some cruise lines have single cruiser get togethers at the start of each cruise, so as you can meet your fellow single travellers. Check to see if the line you are cruising with do this. If you have booked a large dinner table, you will more than likely have another single traveller on it. We had two singles on our table of eight on our last cruise. If you get a chance, try to go to things like dance classes, etc. You are bound to make friends with other that way. Happy cruising.