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(C) The lines are given by $y=x+r$ and $y=-x+r$ for $r \in \{0, 1, 2, 3, 4, 5, 6\}$.These lines form a grid of squares.For a square to have a side len

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$25$

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(C) The lines are given by $y=x+r$ and $y=-x+r$ for $r \in \{0, 1, 2, 3, 4, 5, 6\}$.These lines form a grid of squares.For a square to have a side length of $\sqrt{2}$,the distance between two consecutive parallel lines must be $\sqrt{2}$.The distance between two parallel lines $y=x+r_1$ and $y=x+r_2$ is given by $d = \frac{|r_1 - r_2|}{\sqrt{1^2 + (-1)^2}} = \frac{|r_1 - r_2|}{\sqrt{2}}$.Setting $d = \sqrt{2}$,we get $|r_1 - r_2| = 2$.For the set $r \in \{0, 1, 2, 3, 4, 5, 6\}$,the pairs $(r_1, r_2)$ with a difference of $2$ are $(0, 2), (1, 3), (2, 4), (3, 5), (4, 6)$.There are $5$ such pairs for the lines $y=x+r$ and $5$ such pairs for the lines $y=-x+r$.The number of squares formed is the product of the number of intervals of length $2$ in each direction,which is $5 	imes 5 = 25$.

Consider the fourteen lines in the plane given by $y=x+r$ and $y=-x+r$,where $r \in \{0, 1, 2, 3, 4, 5, 6\}$. The number of squares formed by these lines,whose sides are of length $\sqrt{2}$,is:

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