  The tiling problem has been a famous problem that has appeared in many
Mathematics problems. Many of its solutions are rooted in high-level
Mathematics. Thus we hope to tackle this problem using more elementary
Mathematics concepts. In this report, we start with the simplest cases, with
the smaller numbers: the number of ways to tile a $2 \times n$, $3 \times n$,
$4 \times n$ rectangular board using $2 \times 1$ domino tiles, where the
number of rows is fixed and we present a recursive formula based on $m$ and the
earlier terms. This allows us to deduce the non-recursive formula for each case
that is only dependent on $m$. For each case, we also expand and generalize the
problem, not just for $2$, $3$, $4$ but for any positive integer $k$, for
certain types of configurations of the board. We also focus on one of the
famous variations of the tiling problem: tatami tiling, and present a solution
for simple cases: $2 \times n$, $3 \times n$, $4 \times n$. In the end, we have
managed to find a simpler solution for three different configurations of the
board, with some we even deduced the non-recursive formula. We have also solved
simple cases of the tatami tiling problem, with the hope to tackle the general
case in the future. We realized that our method only works on a case-by-case
basis, with little success in solving the general case. This approach is also
applicable in many other counting problems which we wish to pursue for further
research.
