What is a square root that lies between 17 and 18

You can be as simple-minded as you like. Try 4.14.1, 4.24.2, 4.34.3, etc., squaring each one, till you find one of the squares to lie between 1717 and 1818. In case you jumped over the interval ⟨17,18⟩ ⟨17,18⟩, you could work on varying the second decimal place.