The Game of 24

Date: 14 Mar 1995 12:46:49 -0500
From: Vasha Rosenblum
Subject: math

Dear Dr. Math,

David and I want to know if a problem in the game 
24 is possible?  

Let me give you some background information.  Here are 
the rules. You can only use addition, subtraction, 
multiplication, and division. There can't be  remainders 
like 24r1. The object of the game is to get 24 using all 
four numbers. For example, the numbers 1, 2, 3, 4 would 
be solved in this way: 
 
   4 * 3 = 12 * 2 = 24 * 1 = 24
   
There can be more than one way to solve the problem. So
our question to you is: can you find a way to solve this:
7, 7, 3, 3.  Can you figure it out?  If so, please tell us.

Sincerely,
Greg Rosenbaum and David Mahon

Date: 26 Mar 1995 21:48:55 -0500
From: Dr. Sydney
Subject: Re: math

Hello, Greg and David!  

Thanks for writing.  I'm sorry it has taken us so long to 
respond to your question.  After thinking about it for a 
while, I don't really think there is any way to get 24 out 
of the numbers 7, 7, 3, 3.  Here is why:

It seems to me that division can't be used here because 
we only would be able to divide 7/7 and 3/3 (otherwise 
there would be remainder).  So, whatever the case, we 
would after dividing have only 2 3's or 2 7's and a one to 
work with, and this simply will not work.

So, consider combining addition, subtraction, and 
multiplication.  Can you figure out a combination of these 
operations that will yield 24?  Whatever the combination 
is, it will either have the form ac where a and c are 
integers or it will have the form d + b, where d and b are 
integers.

Let's consider the first case.  What combinations of 2 
integers produce 24 when multiplied together?  Well, 
24 = 24 *1 = 12 * 2 = 8 * 3 = 6 * 4, right?  So, take each 
of these cases individually.  Can we manipulate 2 7's
and 2 3's to get a 24 and a 1?  How about a 12 and a 2?  
a 3 and an 8?, and so on....

If you continue considering cases like this, I think you 
will systematically show that it cannot be done (though, 
I admit I haven't tried all the cases myself - I am leaving 
that up to you!).  I hope this has helped answer your
question.  If you have any more questions, feel free to 
write back.

 -Dr. Sydney, The Geometry Forum

From: Nian Huang
Subject: The Game of 24
Date: Sun, 25 Jan 1998 21:58:45 -0500

Dear Dr. Math,

I searched for "The Game of Make 24" on the Internet and 
found your site.

I found a question dated 1995 about "can you use 7, 7, 3 
and 3 to make 24?" Your answer is wrong! I have a solution 
here.

        (3/7+3)x7 = 24
        
I created a computer game a few years ago which gives the
above solution. Please visit "The Home of Make24" at
        
   http://members.xoom.com/NianQing   

Thanks,

Nianqing Huang