Baltic Way 2019

Answer: $k=2$.

Example. Put $300$ in the low left corner and $101$ in the cell above it.

Estimation. One open card does not determine the snake uniquely, because all the cells of $3\times 100$ board can be considered as one cyclic path: let the rows of the board be denoted by letters a, b, c and columns be numbered from $1$ to $300$, then that path is $$a_1-b_1-c_1-c_2-b_2-b_3-c_3-c_4-b_4-b_5-\cdots -c_{300}-b_{300}-a_{300}-a_{299}-a_{298}-\cdots -a_2-a_1$$ If we open only one card in this path we can orient this path in two ways and put cards such that they increase in the direction of the chosen orientation. So we have at least two configurations here.