CADAEIBFEC and Other NCTM Questions

Date: 10/27/98 at 02:31:14
From: Nick Moreno
Subject: CADAEIBFEC

Hello Dr. Math,

I have three questions from NCTM:

1) The unusual word CADAEIBFEC is a mnemonic for an important piece of 
mathematical information.  What is it?

2) How many digits does the number 25 to the 16th power times 2 to the 
38th power have?

3) On a flat surface, arrange four unit spheres into two layers, with 
three spheres on the lower layer and one sphere on the top layer.  
Find the distance between the highest point of the top sphere and the 
flat surface.

How do you measure the hole made by the three spheres, and then how do 
you measure the arc of the top ball to know how far in it will rest?

There were 31 questions originally, and I'm down to the above three.
If you decide to answer these questions, it would be great. They are 
all from NCTM and probably other students are looking for them as well.  

Either way, thanks! This is a good website to know about.

Nick

Date: 10/27/98 at 08:22:33
From: Doctor Rob
Subject: Re: CADAEIBFEC

1) Hint: Each letter stands for a different digit, forming a constant 
number you should be familiar with. Match the digits by counting the 
letter's place in the alphabet. For example, C is the third letter of 
the alphabet, so replace all C's with a 3.

2) This can be an exercise in the use of logarithms, or you can do it
this way:

   25^16*2^38 = (5^2)^16*2^38
              = 5^32*2^38
              = 5^32*2^32*2^6
              = 2^6*10^32
              = 64*10^32

Now it's easy to count the number of digits.

3) The centers of the four spheres form the vertices of a regular
tetrahedron whose sides have length twice the radius of the spheres.  
The answer you want is the altitude of the tetrahedron, plus the 
distance from the flat surface to the base of the tetrahedron, plus 
the distance from the top vertex of the tetrahedron to the top of the 
top sphere. You can find the tetrahedron at the following URL (follow 
the link to tetrahedron):
  
http://mathforum.org/dr.math/faq/formulas/faq.polyhedron.html   

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