In the PCP, can we remove all dominos with the same top and bottom strings and still get a match?
Intuitively this seems like it would be true at first, but it's because we think too literally about the domino example. Remember in the PCP the strings on the top and bottom of the domino are not necessarily the same length (i.e, the dominos are not necessarily rectangular)
Consider the following example
\begin{align*}
P=\left\{\left(\begin{array}[ccc] \\
1 &amp; &amp; \\
1 &amp; 2 &amp; 3 \\
\end{array}  \right),
\left(\begin{array}[c] \\
2  \\
2  \\
\end{array}  \right),
\left(\begin{array}[cc] \\
3 &amp; 2  \\
4 &amp;    \\
\end{array}  \right),
\left(\begin{array}[cc] \\
4 &amp; 4  \\
4 &amp;   \\
\end{array}  \right)
\right\}
\end{align*}
Here notice there is a match by placing each of the dominos exactly once, and in the order they appear above. On the other had let's try another version.
\begin{align*}
P'=\left\{\left(\begin{array}[ccc] \\
1 &amp; &amp; \\
1 &amp; 2 &amp; 3 \\
\end{array}  \right),
\left(\begin{array}[cc] \\
3 &amp; 2  \\
4 &amp;    \\
\end{array}  \right),
\left(\begin{array}[cc] \\
4 &amp; 4  \\
4 &amp;   \\
\end{array}  \right)
\right\}
\end{align*}
Notice $P'$ was obtained from $P$ by removing $\left(\begin{array}[c] \\
2  \\
2  \\
\end{array}  \right)$, which has the same string on top and bottom.
If there were a match in $P'$ it must start with $\left(\begin{array}[cc] \\
4 &amp; 4  \\
4 &amp;   \\
\end{array}  \right)$ since that is the only time starting and ending with the same letter that can be added to (there is no tile starting with $2$ on top to place after $\left(\begin{array}[ccc] \\
1 &amp; &amp; \\
1 &amp; 2 &amp; 3 \\
\end{array}  \right)$.
This can be followed by either another $\left(\begin{array}[cc] \\
4 &amp; 4  \\
4 &amp;   \\
\end{array}  \right)$, or $\left(\begin{array}[cc] \\
3 &amp; 2  \\
4 &amp;    \\
\end{array}  \right)$. The first is not helpful, and the second places the substring $32$ on the top, which can never be formed on the bottom. So $P'$ does not have a match.