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This paper proposes three ILP-based static thermal-aware mapping algorithms for 3D Networks on Chip (NoC) to explore the thermal constraints and their effects on temperature and performance. Through complexity analysis, we show that the first algorithm, an optimal one, is not suitable for 3D NoC. Therefore, we develop two approximation algorithms and analyze their algorithmic complexities to show their proficiency. As the simulation results show, the mapping algorithms that employ direct thermal calculation to minimize the temperature reduce the peak temperature by up to 24% and 22%, for the benchmarks that have the highest communication rate and largest number of tasks, respectively. This comes at the price of a higher power-delay product. This exploration shows that considering power balancing early in the mapping algorithms does not affect the chip temperature. Moreover, it shows that considering the explicit performance constraint in the thermal mapping has no major effect on performance.