In this paper, it is shown that a polyomino is balanced if and only if it is simple. As a consequence, one obtains that the coordinate ring of a simple polyomino is a normal Cohen–Macaulay domain.
"The coordinate ring of a simple polyomino." Illinois J. Math. 58 (4) 981 - 995, Winter 2014. https://doi.org/10.1215/ijm/1446819295