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

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

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.

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.