This problem in a slightly different wording is known as the {\bf Bulgarian solitaire}. We received three solutions. Prof. Vladislav Kargin submitted a beautiful solution, much simpler than our original solution (which reduces the problem to our Problem 6 this semester). The solutions by Aleksandr Aksenchuk and Krishnaraj Sambath are incomplete and contain some interesting partial observations, like the realization that the equality $5050=1+2+\ldots+100$ plays important role here and that from some day on we should always have one cave with exactly $i$ bats for $i=1,2,\ldots, 100$. For detailed solutions see the following link Solution. See also the following link for a nice discussion of this problem https://www.researchgate.net/publication/273158618_The_Bulgarian_solitaire_and_the_mathematics_around_it