Question

The curator of an art exhibit wants to place security guards along the four walls of a large auditorium so that each wall has the same number of guards. Any guard who is placed in a corner can watch the two adjacent walls, but each of the other guards can watch only the wall by which she or he is placed. There is a maximum of one guard per corner. Draw a sketch to show how this can be done with 6 security guards. Show how this can be done for each of the following numbers of security guards: 7, 8, 9, 10, 11, and 12.

- A) 4 guards per corner, 2 guards along each wall
- B) 3 guards per corner, 2 guards along each wall
- C) 2 guards per corner, 2 guards along each wall
- D) 1 guard per corner, 2 guards along each wall

Answer 1

Security guards must be placed to create equal visibility along four walls, with corner guards counting for both adjacent walls. Only even distributions are possible, making it workable with 6, 8, 10, or 12 guards, but not with 7, 9, or 11 guards.

Step-by-step explanation:

The question involves the placement of security guards in an auditorium in such a way that each wall has the same number of guards, and a maximum of one guard can be placed in each corner.

This is a problem of dividing the guards into four equal groups to guard the four walls, considering that guards in the corners count for both walls they are adjacent to. I'll go through each scenario from 6 to 12 guards and explain the placement.

• With 6 security guards, you would have 1 guard in each corner and 1 guard in the middle of the opposite walls, resulting in 2 guards effectively watching each wall.
• For 7 security guards, this is not possible as it would require having an uneven distribution of guards among the walls.
• With 8 security guards, you would place 1 guard at each corner and 1 guard at each wall center, making 2 guards effectively for each wall.
• For 9 security guards, place 1 guard at each corner and one guard on each wall, except for one wall which will have 2 guards, making it impossible to have the same number of guards on each wall.
• With 10 security guards, place 1 guard at each corner, and then 2 guards along each wall, spaced evenly, except for one wall that only gets 1 additional guard.
• 11 security guards cannot be evenly distributed to ensure each wall has the same number of guards.
• With 12 security guards, place 1 guard at each corner and evenly distribute the remaining 8 guards by placing 2 on each wall.

In summary, 7, 9, and 11 security guards cannot be distributed to ensure each wall has the same number of guards. However, 6, 8, 10, and 12 guards can be placed accordingly.