Fountain Parks

문제

In a nearby park, there are $nn$ fountains, labeled from $00$ to $n-1n\; -\; 1$. We model the fountains as points on a two-dimensional plane. Namely, fountain $ii$ ($0\le i\le n-10\; \backslash le\; i\; \backslash le\; n\; -\; 1$) is a point $(x[i],y[i])(x[i],\; y[i])$ where $x[i]x[i]$ and $y[i]y[i]$ are even integers. The locations of the fountains are all distinct.

Timothy the architect has been hired to plan the construction of some roads and the placement of one bench per road. A road is a horizontal or vertical line segment of length $22$, whose endpoints are two distinct fountains. The roads should be constructed such that one can travel between any two fountains by moving along roads. Initially, there are no roads in the park.

For each road, exactly one bench needs to be placed in the park and assigned to (i.e., face) that road. Each bench must be placed at some point $(a,b)(a,\; b)$ such that $aa$ and $bb$ are odd integers. The locations of the benches must be all distinct. A bench at $(a,b)(a,\; b)$ can only be assigned to a road if both of the road's endpoints are among $(a-1,b-1)(a\; -\; 1,\; b\; -\; 1)$, $(a-1,b+1)(a\; -\; 1,\; b\; +\; 1)$, $(a+1,b-1)(a\; +\; 1,\; b\; -\; 1)$ and $(a+1,b+1)(a\; +\; 1,\; b\; +\; 1)$. For example, the bench at $(3,3)(3,\; 3)$ can only be assigned to a road, which is one of the four line segments (2, 2) – (2, 4), (2, 4) – (4, 4), (4, 4) – (4, 2), (4, 2) – (2, 2).

Help Timothy determine if it is possible to construct roads, and place and assign benches satisfying all conditions given above, and if so, provide him with a feasible solution. If there are multiple feasible solutions that satisfy all conditions, you can report any of them.