Assuming you have completed the map. You cannot solve the cannon puzzles without the map.
Each cannon requires you to input five roman numerals, which you must discover from the map. Match the symbol seen atop the cannon with the same symbol found on the border of the map, and find the roman numeral associated with that symbol in the map.
The puzzle is randomized for each player, so you must read the map symbols in order to solve it. There is no all-out spoiler, sorry.
Hope this helps!!