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Mix the sugar with 180ml hot water in a heatproof bowl until the sugar has dissolved. Leave to cool completely. Stir in the lemon juice.
Tip the cucumbers into a blender with a splash of water and blitz until liquified. Tip some ice, the vodka, the lemonade from step 1 and the puréed cucumbers (see tip, below) into a large pitcher and stir well.
Top up the pitcher with soda water and stir well again, then drop in a few lemon slices just before serving.