In a saucepan, dissolve the sugar in the water by bringing it to a boil over medium-high heat. Turn off the heat and allow it to cool.
Combine the sugar water with the pineapple juice, lemonade, orange juice and pureed strawberries in a 6-quart container. Mix in the rum to taste. Put the entire mixture into a large plastic ice cream pail and freeze overnight.
To make the drinks to order, scoop 1/2 cup of the frozen mixture into an 8 oz glass. Fill the glass with lemon-lime soda and serve with a straw. Keeps well in freezer for up to one month.