In a blender, add chunks of watermelon and puree until you have about 24 oz of watermelon juice.
Fill an ice cube tray with pureed juice, setting aside 12 oz of juice later, and freeze for at least 2½ hours.
Fill a tall glasses with 4 watermelon cubes.
Add pureed watermelon, rum, lime juice, and cucumber juice to a cocktail shaker with a few ice cubes and shake vigorously for 10-15 seconds.
Strain cocktail mixture over the ice in glass.
Top with 1½ oz mint/cucumber/lime mixer
Garnish with a watermelon chunk, cucumber slice, and mint.