The theorem of Fraenkel and Simpson states that the maximum number
of distinct squares that a word w of length n can contain is
less than 2n. This is based on the fact that no more than two
squares can have their last occurrences starting at the same
position. In this paper we show that the maximum number of the last
occurrences of squares per position in a partial word containing one
hole is 2k, where k is the size of the alphabet. Moreover, we
prove that the number of distinct squares in a partial word with one
hole and of length n is less than 4n, regardless of the size of
the alphabet. For binary partial words, this upper bound can be
reduced to 3n.