There's an infinite board. Imagine you add a rectangle of $m*n$ pieces. With $m,n \geq 2$ (There's a piece every square, and you can't put one above other.) You can make a piece 'jump' other that is next to it (Vertically or horizontally only, not diagonally), if the square next to the "jumped" piece (In the same direction) is empty, also, the "jumped" piece disappears. After many movements, What is the least amount of pieces that can be on the board?
So, it's obvious that the answer can't be $0$ since there's no point of it, it has to be at least one if we have, for example, to pieces left of all the $mn$ pieces that are connected each other. I tried to do a coloring of the infinite board, in a chess pattern, so we have Black pieces (Pieces in a black tile) and White pieces, this make any white piece unable to make dissapear another white one and the same with black ones. At the end, the least we have of each color the better, because if we get to have only two left with different colors and connected, then we'll know the least number of pieces is 1. i tried looking for different rectangles of pieces, and i got 1 and 2 as answers. For example doing the following when $m=3$ and $n=4$ makes the board have 1 piece at the end: (From left to right)
It clearly follows that it has 1 piece at the end. I found other cases as ($m=2,n=4$), ($m=2, n=5$), but i also found cases where i got the least is 2, like ($m=2, n=3$), ($m=3, n=3$), etc. i haven't noticed anything else besides this. Any suggestions?