Seats at the Table or No Function Too Large

This puzzle will teach you about mathematical rules with two variables, called functions.

A catering company specializes in dinners for large groups. They advertise: “We can seat your entire party at one table; no function is too large for us.” Here’s how they do it.

They have a large supply of folding tables. Each table seats four people. To make a larger table, they simply place two tables next to each other, then three in a row, and so forth. All the tables are in one row.

How many people can sit at two tables? Five tables? Ten tables?

Can you find a general rule or formula for the number of people, P, who can sit at T tables? Explain how you know your rule is correct.

Background

This is an example of a mathematical function, a rule relating two variables, so that if you know one of the variables, called the independent variable, the rule tells you the value of the other variable, the dependent variable. In this example, T is the independent variable and P is the dependent variable. We say that P is a function of T.

T (tables)	P (people)
1	4
2	6
3
4
5
. . .
10

T

Hint: First, it may help you to make drawings of several situations to help understand the rule. You might want to use graph paper.

Second, a very important step in understanding any function is to make a table showing corresponding values of the variables. It’s easier to read if you put the independent variable on the left, as we have done here.

Course:

Math [1]
Algebra [2]

Result/Solution(s)

Solution: Seats at the Table or "No Function Too Large" Math Puzzle

Here are three different ways that students have arrived at the solution to this problem. All of them started by making drawings.

Maria's Solution

I used my drawings to fill in the table until I could see a pattern. By the time I got to 5, I could see that it increased by 2 every time and could figure out that 22 people could sit at 10 tables.

4 people

6 people

8 people

10 people

T (tables)	P (people)
1	4
2	6
3	8
4	10
5	12
. . .
10

T

So I guessed that I needed to multiply T by 2. But when I tried the rule, P = 2T, I got incorrect results.

Results for P = 2T

T (tables)	P (people)
1	2
2	4
3	6
4	8
5	10
. . .
10	20

T	2P

Now I could see that I would have gotten the correct results if I had added 2 each time. So the correct rule is P = 2T + 2.

Yenching's Solution

I made drawings for 1, 2, and 3 tables. I counted at the tops and bottoms and the ends of the tables.

tables

Then I drew a sketch of a long table, T tables long.

	T seats along top
1 seat at each end		1 seat at each end
	T seats along bottom

I could see that there would be T people sitting along the top, another T people along the bottom, and always 1 person at each end. So I decided that the rule is P = 2 x T + 2. I tested this rule for 1, 2, 3, and 10 tables, so I decided it would always work.

Gupta's Solution

I made a few drawings, and I noticed that every time I added a new table, I added 2 more seats. Even though each new table could seat 4, when I added it to the long table, 2 seats disappeared, the end of the old table plus one side of the new table.

8 seats		4 seats

	2 seats disappear

Since I added exactly 2 seats every time, I figured that I needed to multiply the number of tables by two. Then I remembered that the very first table had seats for 4 people, so there would always be 2 more seats than the number of tables. So my rule is P = 2T + 2. I checked several cases to make sure this always worked.