1. in an honest way and without any doubt We won the match fair and square.
2. if you hit someone fair and square on a particular part of their body, you hit them hard exactly on that part She hit me fair and square on the nose.
Just and honest, as in He won the race fair and square. This redundant expression- fair and square mean essentially the same thing-probably owes its long life to its rhyme. [Early 1600s]