Question

There are five houses on each side of a street, as shown in the figure above. No two houses next to each other on the same side of the street and no two houses directly across from each other on opposite sides of the street can be painted the same color. If the houses labeled $G$ are painted grey, how many of the seven remaining houses cannot be painted grey?

• A. Two
• B. Three
• C. Four
• D. Five
• E. Six

Answer 1

The correct option is E Six

Given

$A\left(G\right)$ $B$ $C$ $D\left(G\right)$ $E$

$STREET$

$F$ $G$ $H\left(G\right)$ $I$ $J$

Let the houses be represented as above,

$A\left(G\right)$, $D\left(G\right)$, $H\left(G\right)$ be the houses painted with gray.

To find the number of houses that cannot be painted gray,

Let us find the number of houses opposite and adjacent ( next to each other on the same side ) to the houses that are painted with gray.

As the houses opposite to each other cannot be painted with the same color.

${}^{\prime }{F}^{\prime }$ is opposite of ${}^{\prime }A(G{)}^{\prime }$

${}^{\prime }{C}^{\prime }$ is opposite of ${}^{\prime }H(G{)}^{\prime }$

${}^{\prime }{I}^{\prime }$ is opposite to ${}^{\prime }D(G{)}^{\prime }$

The houses ${}^{\prime }{C}^{\prime }$, ${}^{\prime }{F}^{\prime }$, ${}^{\prime }{I}^{\prime }$ cannot be painted gray.

As the houses adjacent to each other cannot be painted with the same color,

${}^{\prime }{B}^{\prime }$ is adjacent to ${}^{\prime }A(G{)}^{\prime }$

${}^{\prime }{G}^{\prime }$ and ${}^{\prime }{I}^{\prime }$ are adjacent to ${}^{\prime }H(G{)}^{\prime }$

${}^{\prime }{C}^{\prime }$ and ${}^{\prime }{E}^{\prime }$ are adjacent to ${}^{\prime }D(G{)}^{\prime }$

The houses ${}^{\prime }{C}^{\prime }$, ${}^{\prime }{E}^{\prime }$, ${}^{\prime }{B}^{\prime }$, ${}^{\prime }{G}^{\prime }$, ${}^{\prime }{I}^{\prime }$ cannot be painted gray.

Hence,

The houses ${}^{\prime }{B}^{\prime },$ ${}^{\prime }{C}^{\prime },$ ${}^{\prime }{E}^{\prime },$ ${}^{\prime }{F}^{\prime },$ ${}^{\prime }{G}^{\prime },$ ${}^{\prime }{I}^{\prime }$ cannot be painted gray.

Therefore, Number of houses among seven remaining houses that cannot be painted gray are ${}^{\prime }{6}^{\prime }$.