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### 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
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

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
```
Associated Topics:
Elementary Puzzles

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