# Puzzle | Six Houses P, Q, R, S, T, and U

There are three houses on each side of the road. These six houses are labeled as **P**, **Q**, **R**, **S**, **T**, and **U**. The houses are of different colors, namely, **Red**, **Blue**, **Green**, **Orange**, **Yellow**, and **White**. The houses are of different heights.

- T, the tallest house, is exactly opposite to the Red-colored house.
- The shortest house is exactly opposite to the Green colored house.
- U, the Orange colored house, is located between P and S.
- R, the Yellow-colored house, is exactly opposite to P.
- Q, the Green colored house, is exactly opposite to U.
- P, the white-colored house, is taller than R, but shorter than S and Q.

What is the **color of the tallest house** and how many houses are taller than the house **R**?

__Solution:__

- From the given information the tallest house is
**T**and it is opposite to the Red-colored house. and the Shortest house is exactly opposite to the**Green**colored house. - The color of the house
**U**is**Orange**and it is situated**P**and**S**. It implies that houses**P**,**S**, and**U**are on one side of the road and the other three houses, namely,**Q**,**R**, and**T**are on another side of the road.**P**,**S**, and**U**can be placed in two ways. One way is**{P, U, S}**, and another way is**{S, U, P}**. - The color of the house
**R**is**Yellow**and it is opposite to**P**. - The color of the house
**Q**is**Green**and it is opposite of**U**. The shortest house is exactly opposite to the**Green**colored house it means the shortest house is**U**. - The remaining house
**T**, the tallest house, is opposite to**S**. The color of the house which is exactly opposite to**T**is**Red**. It means the color of**S**is**Red**.**P**, the white-colored house, is taller than**R**, but shorter than**S**and**Q**. The heights of**P**,**Q**,**R**, and**S**can be arranged in two ways. One of the ways is**{S, Q, P, R}**and another way is**{Q, S, P, R}**. - Therefore,
**{P, U, S}**are one side on the road, and**{R, Q, T}**are on another side of the road.**P**is opposite to**R**,**U**is opposite to**Q**and**S**is opposite to**T**. These two cases are shown below:

Heights of houses also have two ways which are shown in the table below:

CASE 1 | CASE 2 | |

Tallest | T-Blue | T-Blue |

2nd Tallest | Q-Green | S-Red |

3rd Tallest | S-Red | Q-Green |

4th Tallest | P-White | P-White |

5th Tallest | R-Yellow | R-Yellow |

Smallest | U-Orange | U-Orange |

From the above table,

The color of the tallest house is **Blue **and the tallest house is **T**. Therefore, there are four houses taller than **R **which are **T**,** Q**, **S**, and **P**.