Consider the first quadrant in the Cartesian plane divided into unit squares by horizontal and vertical lines at the positive integers. (1) Place $latex 3$ dots (clones) in the shape of an $latex L$-tromino in the bottom left-most squares, and…

Let $latex R$ be a rectangle with side lengths $latex a$ and $latex b$. Show that if $latex R$ is tiled by rectangles each of which has at least one integer side, then at least one of $latex a$ and…

Define a sequence $latex a_0$, $latex a_1$, $latex a_2$, $latex \ldots$ of integers as follows: $latex a_0=0$, and given $latex a_0$, $latex a_1$, $latex a_2$, $latex \ldots$, $latex a_n$, then $latex a_{n+1}$ is the least integer greater than $latex a_n$…