The paper deals with the so called hungry periodic discrete (hpd) Toda equations described by \[ I_n^{t+M} = I_n^t + V_n^t-V_{n-1}^{t+1}\;,\;V_n^{t+1} = {{I_{n+1}^tV_n^t} \over{I_n^{t+M}}} \] with the periodic boundary conditions \[ I_n^t = I^t_{n+N}\;,\;V_n^t = V^t_{n+N} \] and \(N\), \(M\) positive integers, \(t\) is the time variable and \(n\) is the position. The following assumption is made \[ \prod_{n=1}^N I_n^{t+M} = \prod_{n=1}^N I_n^t \neq \prod_{n=1}^N V_n^{t+1} = \prod_{n=1}^N V_n^t \] to ensure the existence of a unique solution. This solution is constructed using the auxiliary \(\tau\)-(tau)-function.