Determining the value of Tn with a board and bricks

Question

I have this homework question and I'm not sure where to start or how to do either of the problems at the bottom of the question. Any help appreciated!

Answer 1

Because the board is always 5 units high, you have to stack one horizontal and one vertical brick on each other to fill the height, without loss of generality we will assume the horizontal one is on the bottom:

xx
xx
xx.
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Now what can you place right of the bottom one? No vertical brick can go there, because it would leave the position with the dot (.) unfillable. So we have

xx
xx
xx
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Now right of the vertical brick, you cannot place a horizontal brick, because it would leave you with a $1\times3$ block either above or below your horizontal brick that cannot be filled. Thus, you have to put another vertical brick there, and yet another one besides that one. And by only placing one brick, you were forced to fill up a $B_6$.

And you had only one choice: Put the horizontal brick on top or at the bottom. Thus, $T_6=2$.

Now that answers both questions at once: You cannot fill a $B_n$ where $n$ is not a multiple of $6$, because we were forced to fill 6 columns at once in order to not leave any blanks, and when you fill a multiple of 6 columns, you have one choice to make every 6 columns, so $T_{6k}=2^k$.