Enigma 1326: Roman squares
From New Scientist #2485, 5th February 2005
Harry and Tom each chose two consecutive integers between 1 and 20 and wrote down their squares as Roman numerals. I challenged each of them independently to replace each letter of his Roman numerals with a digit (with any given letter always replaced by the same digit, different letters by different digits and leading zeroes not permitted) in such a way that the numbers they created from their Roman squares were themselves consecutive perfect squares.
I leave you to discover the connections between their solutions, but will add that from their ascending Roman squares Harry created descending consecutive numerical squares and Tom created ascending consecutive numerical squares. Submit both the two Roman squares and the two numerical squares created from them that (a) Harry and (b) Tom came up with.