1. Place 2 cups water and chopped cucumber in a blender; process until smooth. Strain cucumber mixture through a sieve into a medium bowl; discard solids.
2. Place 1 tablespoon sugar, 1 tablespoon lime juice, and 4 mint leaves in a 2-cup glass measure; crush with a wooden spoon. Add 3 tablespoons rum and about 1/2 cup cucumber mixture; stir until sugar dissolves. Stir in 1/4 cup sparkling water. Place 1/4 cup ice in a 12-ounce glass; pour cucumber mixture over ice. Garnish with 1 mint sprig, 1 cucumber slice, and 1 lime wedge. Repeat procedure with remaining ingredients. Serve immediately.
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