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
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