The well-known Knaster conjecture claims the following: for any points \(w_1,\dots, w_k\in S^{k+m-2}\) and an arbitrary (continuous map) \(f: S^{k+m-2}\to \mathbb{R}^m\) there is a rotation \(A\in \text{SO}(k+m-1)\) of the sphere such that \(f(Aw_1)=\cdots= f(Aw_k)\). The author proves this conjecture for the case in which \(n= p^2\) for an odd prime \(p\) and the points lie on a circle and divide it into equal parts.