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Difficulty: HardSeating Arrangement

Eight officials (Auditor, Clerk, Governor, Judge, Mayor, Senator, Sheriff, and Treasurer) are seated around a square table. Four of them sit at the corners facing the center of the table, while the other four sit in the middle of each side facing outward away from the center.

- The Sheriff sits at one of the corners.
- The Governor sits third to the right of the Sheriff.
- The Mayor sits second to the right of the Governor.
- The Senator and the Judge are immediate neighbors, but neither is an immediate neighbor of the Governor or the Sheriff.
- The Treasurer sits third to the right of the Judge.
- The Clerk is not an immediate neighbor of the Governor.

Based on the given seating arrangement, which of the following statements are correct?

  1. The Clerk and the Governor sit exactly opposite each other.Answer
  2. B
    The Auditor sits second to the left of the Treasurer.
  3. Exactly three people sit between the Judge and the Mayor.Answer
  4. D
    The Senator sits to the immediate left of the Clerk.

Answer

The statements indicating that the Clerk and Governor sit opposite each other, and that exactly three people sit between the Judge and the Mayor, are correct.
Based on the systematic deduction of all constraints, the table is arranged as follows (clockwise from top-left): Sheriff, Clerk, Senator, Judge, Auditor, Governor, Treasurer, and Mayor. In this specific configuration, the Clerk and Governor occupy directly opposite middle seats, and the Judge and Mayor occupy opposite middle seats, meaning there are exactly three people between them on either side.

Step-by-Step Solution

1
Establish the directional rules for the inward and outward facing seats.
For the 4 corners facing inward, 'right' moves counter-clockwise and 'left' moves clockwise. For the 4 middle positions facing outward, 'right' moves clockwise and 'left' moves counter-clockwise.
Correctly identifying left and right based on facing direction is mandatory before placing any individuals.
2
Place the Sheriff, Governor, and Mayor to create a foundational layout.
Place the Sheriff at a corner (e.g., top-left, inward). Third to the right (counter-clockwise) places the Governor at the bottom-middle position (outward). Second to the right of the Governor (clockwise) places the Mayor at the left-middle position (outward).
These three linked conditions provide fixed anchor points on the table.
3
Determine the available positions for the Senator and the Judge.
They must sit in adjacent seats but cannot sit next to the Governor or the Sheriff. Eliminating the seats adjacent to the Governor and Sheriff leaves only the top-right corner and the right-middle position as the single valid pair of adjacent seats.
Negative constraints drastically reduce the possible locations for grouped individuals.
4
Differentiate the specific seats for the Senator and the Judge using the Treasurer's condition.
The Treasurer is third to the right of the Judge. If the Judge were at the top-right corner (inward), third to the right would be the left-middle seat, which is already occupied by the Mayor. Thus, the Judge must be at the right-middle seat (outward). Third to the right (clockwise) places the Treasurer at the bottom-left corner. The Senator takes the top-right corner.
Testing the directional constraint ensures no two people are assigned to the same seat.
5
Place the remaining officials, the Clerk and the Auditor.
The remaining empty seats are the top-middle and bottom-right corner. The Clerk cannot be a neighbor of the Governor, so the Clerk takes the top-middle seat. The Auditor fills the final seat at the bottom-right corner.
Applying the final negative constraint finalizes the complete seating arrangement.

Key Concept

Solving complex seating arrangements with mixed facing directions by systematically tracking relative left/right movements.
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Topics

Seating Arrangement

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