Pyramidal Numbers

Date: 07/04/2001 at 14:21:07
From: karinny
Subject: Pyramid

Can I make a square pyramid with 1000 tennis ball?

Date: 07/05/2001 at 12:50:09
From: Doctor Achilles
Subject: Re: Pyramid

Hi Karinny,

Thanks for writing to Dr. Math.

Let's start out by looking at building triangles (looking at them from 
the side) with tennis balls; then we'll move on to pyramids.

The first "triangular number" is 1, just a lone tennis ball:

                       o

The next is 3, so put two balls under the first one:

                       o
                      o o

The next is 6; put three balls under that:

                       o
                      o o
                     o o o

The next is 10; put four balls under that:

                       o
                      o o
                     o o o
                    o o o o

And so on.

So the "triangular numbers" are 1, 3, 6, 10, 15, 21, ...

The formula for "triangular numbers" is:

  1, 1+2, 1+2+3, 1+2+3+4, 1+2+3+4+5, 1+2+3+4+5+6, etc.

It's harder to draw pyramids, but I'll describe them as best I can 
(looking down from the top).

The first "pyramidal number" is one, just a lone tennis ball:

                             o

The next is 5. You get that by putting four balls next to each other 
on the ground, and resting the fifth ball on top of them so it looks 
like a pyramid:

   Bottom row:
                            o o
                            o o

   Top row:
                             o

The next is 14. Put nine balls in a square, then put the next four 
balls on top of that, then put the last ball on top of the four:

   First row:
                           o o o
                           o o o
                           o o o

   Second row:
                            o o
                            o o

   Third row:
                             o

The next is 30. Put 16 balls in a square underneath the 14-ball 
structure you just built:

   First row:
                          o o o o
                          o o o o
                          o o o o
                          o o o o

   Second row:
                           o o o
                           o o o
                           o o o

   Third row:
                            o o
                            o o

   Fourth row:
                             o

And so on.

So the "pyramidal numbers" are 1, 5, 14, 30, 55, 91, ...

The formula for "pyramidal numbers" is (where n^2 means "n squared"):

  1^2, 1^2+2^2, 1^2+2^2+3^2, 1^2+2^2+3^2+4^2, 1^2+2^2+3^2+4^2+5^2, 
etc.

So just keep going with that until you get to 1000. If 1000 is a 
pyramidal number, then you can make a square pyramid out of 1000 
tennis balls.

Hope this helps. If you have any more questions about this or other 
math topics, please write back.

- Doctor Achilles, The Math Forum
  http://mathforum.org/dr.math/