Elementary POW, April 21-25, 1997


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If you build a four-sided pyramid, not counting the bottom as a side, using ping-pong balls, how many balls will there be in a pyramid that has seven layers?

Do you see a pattern in the number of balls necessary to make each layer? (Hint: Start with layer 1 being the one ping-pong ball needed for the top layer. How many balls would be under that ball to make the second layer? How many under those to make the third layer, etc.?)

If you've figured out the pattern, you should be able to tell me how many ping-pong balls there are in the 20th layer.

This week's mentor is Lauren Hillman, Grade 10, Germantown Academy, Fort Washington, PA. Lauren is an active participant in the Geometry Problem of the Week and Project of the Month. She is also very involved in theater productions both at school and in an outside theater group. She plans to be an actress and already has her stage name picked out, Cher Harston.
There were a lot of solutions sent in for this problem. Some students were close in their answer to how many ping pong balls were in the first seven layers (correct answer 140), but others were way off, submitting answers in the millions. Most of the students who correctly solved the problem made some type of model to help them visualize it. Several well thought out solutions are included.

Correct Solutions submitted by:

Highlighted Solutions

Mrs. Gleichman's Sixth Grade Class
mrbockus@hotmail.com
Grace Russell Elementary School,
Wilburton, OK USA

Since we did not have ping pong balls we used small blocks and built the pyramid described. As we built it we recorded the number of blocks needed for each level. Our data are listed below:

The total number needed for a 7 layer pyramid will be 140 ping pong balls. We then noticed that each level's number of balls was the square of the number of its layer. That means that the 20th layer would have 20 x 20 = 400 ping pong balls.

Sierra Seip
5th grade
Phoenix Country Day School

My computer teacher, Mr. Butzel showed me the problems you had posted several weeks ago and challenged me to solve some of them. Here is a solution to the pyramid problem.

I started by building the first 3 layers with some balls from my pool table. Then with the pattern I discovered, I drew a picture. The pattern is: number of balls in level 1 = 1 squared, number of balls in level 2 = 2 squared, number of balls in level 3 = 3 squared, and so on. Then I added up how many balls were in each level to find the sum of all the balls.

For the first part of the problem, there were 140 balls. As for how many balls there would be in the twentieth layer, there would be 20 squared which equals 400.

Andrew Dang, grade 5
Western Salisbury School
(Mrs. Geschel)
Allentown, PA

I first looked at the top layer, second layer, and third layer. The first had 1 ball, the second 4, the third 9. I saw a pattern: 1+3=4, 4+5=9. The first time adds 3, then it adds 5. So each time the number to add to the layer increases by 2.

Then I added the 7 layers and got 140 balls.

To continue:

The 20th layer = 400.

Nathan Strauss, Brian Powers, and Ben Clements
Forsyth School
St Louis, MO
Bob Coulter's third grade math class

If the top layer of ping-pong balls is one,the number needed to support it would be four, and the number needed to support that would be nine, then 16, then 25, then 36, and then 49. The pattern is each layer is the square number for that layer. For example, since 4 multiplied by 4 equals 16, there are 16 balls on the fourth layer. All the balls on the first 7 layers = 140. On the twentieth layer there are 400 balls because 20 times 20 = 400.

Casey Gorish
6th Grade
Mrs. McMahon
Murray Middle School
Ridgecrest, CA

  1. If you build a four-sided pyramid, not counting the bottom as a side, using ping-pong balls, how many balls will there be in a pyramid that has seven layers? 140.

  2. If you figured out the pattern, you should be able to tell me how many ping-pong balls there are in the 20th layer. 400.

    I found this answer by stacking up some marbles. First, I started with a four by four base, then I found that a three by three layer fit on the base and a two by two layer fit on top of that. This led me to the fact that each layer is the layer number squared. For the first question, I added up all the layers. I found that the answer was 140 ping pong balls. For the second question, I multiplied 20 by 20 and found that there are 400 ping pong balls on the 20th layer.

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