<p>This paper investigates the discrete-time nonlinear zero-sum game problem with alternating dual-impulse control based on adaptive dynamic programming (ADP). Such problems appear in scenarios with scheduled discrete interventions, such as nonlinear oscillators with impulsive excitations or periodic medical dosing. The main objective is to determine equilibrium strategies and corresponding saddle-point solutions in impulsive nonlinear systems. The key challenge lies in handling nonsmooth state transitions and the computational complexity of nonlinear dynamics. To overcome these difficulties, we design a periodic cumulative utility function that focuses computation on impulse instants and develop ADP algorithms for efficient learning. Convergence to saddle-point equilibria is theoretically proven when such equilibria exist, and the approach remains effective even when exact equilibria do not exist. Simulations on Duffing and torsional systems confirm both the accuracy and robustness of the proposed framework.</p>