Let Sn denote the staircase board with 1+2+⋯+n=n(n+1) / 2 squares. For example, S4 is × ×

step 1 Okay, so for this question, we want to prove that we can tile a checkboard with an even number o

Let Sn denote the staircase board with 1+2+⋯+n=n(n+1) / 2...

Let $S_{n}$ denote the staircase board with $1+2+\cdots+n=n(n+1) / 2$ squares. For example, $S_{4}$ is $$ \begin{array}{|l|l|l|l|} \hline & \times & \times & \times \\ \hline & & \times & \times \\ \hline & & & \times \\ \hline & & & \\ \hline \end{array} $$ Prove that $S_{n}$ does not have a perfect cover with dominoes for any $n \geq 1$.

Let $S_{n}$ denote the staircase board with $1+2+\cdots+n=n(n+1) / 2$ squares. For example, $S_{4}$ is
$$
\begin{array}{|l|l|l|l|}
\hline & \times & \times & \times \\
\hline & & \times & \times \\
\hline & & & \times \\
\hline & & & \\
\hline
\end{array}
$$
Prove that $S_{n}$ does not have a perfect cover with dominoes for any $n \geq 1$.

Key Concepts

Domino Tiling

Domino tiling is the concept of covering a board, usually a grid or some structured set of squares, with dominoes where each domino covers exactly two adjacent squares. In problems of perfect coverings, the challenge is to determine if it is possible to partition all the squares of a given board completely with dominoes without overlapping or leaving any gaps, often requiring the board's total number of squares to be even.

Parity Argument

A parity argument involves analyzing whether a set of objects can be completely paired based on their even or odd count properties. In tiling problems, since each domino covers two squares, the total number of squares must be even; any imbalance due to the structure of the board or coloring constraints could prevent a complete domino covering. This method is often used to show that, even when the total number of squares might appear acceptable, there can still be an inherent impossibility in covering exactly due to parity mismatches.

Checkerboard Coloring

Checkerboard coloring is a technique where a board is colored in alternating colors (like black and white) to reveal hidden parity or balance issues. Since a domino placed on a checkerboard will always cover one square of each color, counting the number of squares of each color provides an invariant: if the board does not have an equal number of each color, a perfect domino covering is impossible. This method is particularly useful in exposing tiling impossibilities in irregularly shaped boards.

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