Let n is an element of N and let K be a field with a henselian discrete valuation of rank n with hereditarily euclidean residue field. Let F/K be a function field in one variable. It is known that every sum of squares is a sum of 3 squares. We determine the order of the group of nonzero sums of 3 squares modulo sums of 2 squares in F in terms of equivalence classes of certain discrete valuations of F of rank at most n . In the case of function fields of hyperelliptic curves of genus g , K.J. Becher and J. Van Geel showed that the order of this quotient group is bounded by 2 n( g+1) . We show that this bound is optimal. Moreover, in the case where n = 1, we show that if F/K is a hyperelliptic function field such that the order of this quotient group is 2 g + 1 , then F is nonreal. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.