<s>[INST] Can you add a tile to a $7\times 7$ checkerboard so that it is tile-able by $5$ ominoes ($5\times1$ rectangles)? Can you add a tile to a $7\times7$ checkerboard so that it is tile-able by $5$ ominoes ($5\times1$ rectangles)? If so, where must you place the extra tile? If not, explain why.
you can add a tile because you would need the checkerboard to be a multiple of $5$. and a $7\times7$ checkboard means there are $49$ squares which is not a multiple of $5$ so by adding $1$, we get 50 which is a multiple of $50$.
Now im having difficult of where to place the extra tile
 [/INST] Hint:
$$\begin{array}{ccccc} 1 & 2 &3&4&5&1&2\\2&3&4&5&1&2&3\\
3&4&5&1&2&3&4\\ 4&5&1&2&3&4&5\\
5&1&2&3&4&5&1\\ 1&2&3&4&5&1&2\\ 2&3&4&5&1&2&3 \end{array}$$
Each $1\times 5$ tile will cover a $1$, a $2$, a $3$, a $4$, and a $5$.  Is there a tile you can add, within this numbering scheme, and get everything to work out?
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