Ray is setting the table for a birthday diner.He needs to set 12 places at a round table.he has 3 different kinds of plates:white plates; blue plates; and gold plates.How can he set the table so that no two of the same kind of plates are next to each other?
Thank you for your help
white, blue, gold, white, blue, gold, white, blue, gold, white, blue, gold
To solve this problem, we can use a principle called the Pigeonhole Principle.
The Pigeonhole Principle states that if you have more pigeons than pigeonholes, at least two pigeons must go in the same pigeonhole. In this case, the "pigeons" are the plates and the "pigeonholes" are the places at the table.
Since Ray has 12 places at the table and 3 different kinds of plates (white, blue, and gold), we can consider each kind of plate as a pigeonhole. To ensure that no two plates of the same kind are next to each other, we need to distribute the plates in such a way that each pigeonhole (kind of plate) has at most one plate.
Here's one possible way Ray can set the table:
1. Start by placing a plate of any kind at a random place at the table.
2. Alternate between the remaining two kinds of plates, placing one in every other spot around the table. This ensures that no two plates of the same kind are adjacent to each other.
For example, Ray can proceed as follows:
- Place a white plate at the first spot.
- Then, place a blue plate at the third spot.
- Next, place a gold plate at the fifth spot.
- Continue this alternating pattern until all 12 spots are filled.
This way, Ray can set the table so that no two plates of the same kind are next to each other.