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Suppose such a labeling exists. Let $F_k$ the region formed by the cells labeled ${1,2,\dots,k}$. There is a minimum $k>1$ such that $F_k$ connects two opposite sides of the checkerboard (wlog suppose left and right side). Color in black the cells belonging to $F_{k-1}$, and in white the rest.

Then, $k$ is in a white cell touching a black cell, and in every other column there is at least one black and one white cell. So overall there are at least $n$ white cells touching black cells (one per column). One of them must contain a number $\geq k-1+n$, and (since it touches the black region) it touches a number $\leq k-1$. Contradiction!

Post Undeleted by Federico Poloni
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Suppose such a labeling exists. Then there is a minimum Let $k$ such that F_k$the region formed by the cells labeled$1,2,\dots,k$touches two opposite borders of the checkerboard. Without loss of generality, we may assume {1,2,\dots,k}$. There is a minimum $k>1$ such that $k$ touches the topmost border F_k$connects two opposite sides of the checkerboard (otherwise, rotate the checkerboard)wlog suppose left and right side). Let us color Color in black the cells labeled belonging to$1,2,\dots,k-1$. F_{k-1}$, and in white the rest.

Then, $k$ is in a white cell touching a black cell, and in every row other column there is at least one white cell adjacent to a black and one white cell, so . So overall there are at least $n$ overallwhite cells touching black cells. At least one One of these cells them must contain a number $\geq k-1+n$, and (since it is adjacent to touches the black region) it is adjacent to touches a number $\leq k-1$. Contradiction!

Post Deleted by Federico Poloni
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