1. **Problem statement:** We have an infinite grid with ants starting at the lower left corner at time $t=0$. Each second, one ant can duplicate itself and move one copy up and one copy right, but only if both target cells are vacant. The question is whether there exists a time $t$ such that the ants have completely vacated the yellow L-shaped region of 5 cells. 2. **Understanding the yellow region:** The yellow region consists of 5 cells forming an L-shape: - First column (from bottom): 3 cells - Second column (from bottom): 2 cells 3. **Key insight:** Each duplication moves ants up and right, expanding their coverage diagonally. The ants start at the bottom-left cell of the yellow region. 4. **Movement and duplication:** At each time step, ants can only move if both the cell above and the cell to the right are empty. This means ants spread outwards, filling cells diagonally. 5. **Can ants vacate the yellow region?** To vacate a cell, ants must move out of it. But since ants only move by duplicating into adjacent cells up and right, and the yellow region is at the bottom-left corner, ants cannot move left or down to leave the yellow region. 6. **Conclusion:** The ants cannot vacate the yellow region because they start inside it and can only move up and right, which means they will always occupy at least some cells in the yellow region. Therefore, there is **no time $t$** such that the yellow region is completely vacated. **Final answer:** No, there is no time $t$ at which the ants have completely vacated the yellow region.