Four Four's, 2-10

Date: 11/24/97 at 11:48:42
From: Zachary Burns
Subject: Numerical Expression

The numerical expression  4 + 4/4 - 4  uses four 4s and has a value 
of 1. Find a numerical expression using only four 4s for each integer 
from 2 to 10. You may use any operations and grouping symbols.

I do not know where to start on this problem. Can you please help me 
out?

Thanks,
Zachary

Date: 11/24/97 at 13:52:11
From: Doctor Rob
Subject: Re: Numerical Expression

I would start out by figuring out what operations I felt comfortable 
using. Certainly +, -, *, and / should be allowed, and powers.  What
about the square root operation?  What about decimal points (like 
4.4)? What about adjunction (like 444)?  And so on ....  Some of the 
operations require only one 4, like Sqrt[4]. These are called "unary 
operations" (pronounced "YEW-nair-y").  Others require two 4's, like 
4/4.  These are called "binary operations" (pronounced "BYE-nair-y").

I would take one four and apply all the unary operations to make a 
list of values that result:  4, .4, Sqrt[4] = 2, -4, and so on. Then I 
would apply all the unary operators again, to expand the list, then 
again, and again. If we continued this indefinitely, we would have a 
list of all numbers expressible with one 4, such as -Sqrt[-Sqrt[.4]].  
Duplicates can be deleted, such as -(-4) = 4.

Then I would combine these using all the binary operations and all 
numbers on the list to include numbers representable with two 4's, 
such as  Sqrt[4]^(-4). Then I would apply all the unary operators, 
perhaps several times, to these results. This produces "all" numbers 
representable with two 4's.  Part of the list for two 4's would 
contain:

   4 + 4 = 8,
   4 - 4 = 0,
   4 * 4 = 16,
   4 / 4 = 1,
     4^4 = 256,
  4^(-4) = 1/256
   44,
   4.4,
   .44,
  .4 / 4 = 1/10,
   Sqrt[4]^4 = 16 (duplicate of 4*4 = 16),
   Sqrt[4]^(-Sqrt[4]) = 1/4,

and so on.

If you continue this process until you have four 4's, you will have a 
list of "all" numbers representable with four 4's. Most of them will 
not be integers from 0 to 10, so you can discard them.

If you are clever, you can start with the number you want to write and
combine it with two 4's to create a number on your two 4's list.  For
example, let's do 11.  Now 11 and 44 are easily combined:

   11/44 = 1/4 = Sqrt[4]^(-Sqrt[4]),

the last from the list above, so

   11 = 44*Sqrt[4]^(-Sqrt[4]).

Understand?

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/