Counter is one of the basic parts of a digital device. Traditional counters based on Complementary Metal Oxide Semiconductor (CMOS) cannot maintain the data after power down. It may require the extra power and time consumption to store the counting results in memory. Using non-volatile memory to construct the counter can solve these problems. In this paper, we design a 4-bit up-down counter by using the Memristor-Aided LoGIC (MAGIC) and auxiliary CMOS circuits. Furthermore, we verify the design through LTSPICE simulations. The simulation results show that the proposed counter operate correctly, that converts the counting result into a binary number, and stores them in the memristors. Compared with CMOS-based counters, this design has great advantages on area, power and nonvolatility.