In a saucepan, bring sugar and water to a boil. Remove from heat, add mint leaves, close the lid and leave for 10 minutes. Strain in a bowl and refrigerate.
With a fruit knife, cut the zest from the grapefruit into long strips. Put in a bowl and fill with rum. Cover and leave for 30 minutes. Strain.
Carefully cut segments from grapefruit, removing membranes.
In each glass, put 2-3 grapefruit segments, add 0.75 glasses of ice. Separately, mix 3 tablespoons of rum, 2 tablespoons of mint syrup and 0.25 cups of mineral water, pour into a glass. Garnish with mint leaves.