Set aside 4 of the cucumber slices, and put the rest of the slices into the bowl.

Scrub the lime and cut it into quarters from end to end (lengthwise). Put each quarter on the cutting board, flat side down. Cut one quarter into 4 pieces and set aside. Cut each of the remaining quarters into thin slices and add them to the bowl of cucumber slices.

Add the honey or maple syrup and a pinch of salt to the cucumber and lime slices and use the wooden spoon or potato masher to mash them really well until the lime gives up its juice, the cucumber is completely smashed, and the honey or maple syrup is dissolved, about 2 minutes. Add the water and stir well.

Set the strainer over the pitcher and pour the limeade mixture through it. Use the wooden spoon to mash the mixture against the mesh of the strainer to release as much liquid as possible. Serve right away with ice, garnished with the reserved cucumber and lime slices.