$k = 2$ works with these turned over cards: 001 ___ ___ ___ ... ___ ___ ___ ___ ___ ... ___ 199 ___ ___ ___ ... ___ The unique number assignment is: 001 002 003 004 ... 098 099 300 198 197 196 195 ... 101 100 299 199 200 201 202 ... 296 297 298 The reason it's unique is that the 001 -> 199 route can't use the top right corner, which is too far away. 199 -> 300 can just barely reach that corner, and it has to use the bottom right to leave enough room for 001 -> 199. To show $k = 1$ is impossible, take some Hamiltonian cycle on the grid through adjacent squares. Following that cycle in both directions gives at least two ways to put any number on any space.