Suppose you own a catering business. As part of your services, you not only prepare the food, but you also set up tables and chairs. Assume one person can sit comfortably at the edge of one table. You need to set up for 18 people. Since your tablecloths are large, tables must meet along at least one edge. Also, you cannot have single tables that seat four people, because they would be crowded.
How many tables will you need? Are there other possible table arrangements that would require the fewest number of tables? The greatest number of tables? If so, is there any special significance to these solutions? What happens to the perimeter when one table is added? (Make sure to consider all cases.) What, in mathematical terms, do the number of people and the number of tables represent?
If you have not already done so, put your data in a table. Graph various combinations of the data, identifying the various relationships.
Submit your idea for an investigation to InterMath.