Given a table with 3 fields for people, where each person can be either Curly, Lary or Moe, and 3 fields for chairs, where each chair can be either Blue, Red or Green. Here are two such records:

Per1 Per2 Per3 Per1_Chair Per2_Chair Per3_Chair
Curly Lary Moe Blue Green Red
Lary Curly Moe Red Blue Green

How do I get 3 new fields - Blue, Green and Red - each of which shows the person in that color for the respective record:

Blue Green Red
---- ------ -----
Curly Lary Moe
Curly Moe Lary

I did this in Excel by adding 3 columns and using a formula with functions match, offset and lookup, but I want to do this instead with a query.
Thanks in advance.

Your data structure is working against you here. Try reading up on Data Normalization. It will make your life easier.

The table is already given and cannot be changed.
The question is how I can solve the problem using a query and joins using multiple copies of the table.

Ask the professor.