Using a blender or food processor, puree ¾ of the watermelon and cucumber together. Strain the mixture using a fine mesh sieve. Chill.
Juice the limes and muddle the mint using a wooden spoon or a muddler. Put the mixture in a shaker with ice and shake for 15-20 seconds. Strain into a large pitcher.
Add the watermelon/cucumber juice to the pitcher, then the rum or vodka and the sparkling water.
Pour into serving glasses filled with ice and garnish with a slice of watermelon, a couple of slices of cucumber and a sprig of mint.
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