Write a Sum That Totals 100 Using Digits 1 - 9
Date: 05/30/2004 at 00:31:40 From: Claire Subject: Digits 1-9 totaling 100 Is it possible to arrange the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 so that when added they total 100? Only adding is permitted and the numbers can be rearranged. Each number can only be used once. I've been trying for ages to figure this one out and am having a great deal of trouble--I keep getting 99 and 108. Can it be done?
Date: 05/30/2004 at 21:02:58 From: Doctor Vogler Subject: Re: Digits 1-9 totaling 100 Hi Claire, Thanks for writing to Dr Math. I gather that you have to use every digit. If you don't have to use all of them, then it's pretty easy. So I will assume that you have to use them all. In that case, the answer to your question is no, it is not possible. I will try to explain why. First of all, you clearly can't use numbers with three digits or more, because even one such number is already too much. So every number in the sum has to be either a one-digit number or a two-digit number. Second, if you take a digit used on the tens place, and you move it to the tens place of a different number, then that doesn't change anything. For example, 12+3+4+57+6+8+9=99 2+13+4+57+6+8+9=99 2+13+4+7+6+58+9=99 52+3+4+17+6+8+9=99 etc. So really the only question we have to answer is: Which digits go in the tens place, and which digits go in the ones place? Now let's look at the sum like this: 1*(sum of digits in ones place) + 10*(sum of digits in tens place) Now if we split up the second part into two pieces, according to 10 = 9 + 1, then we can look at the sum as 1*(sum of ALL digits) + 9*(sum of digits in tens place). Well, I expect you can figure out that the sum of all of the digits is 1+2+3+4+5+6+7+8+9 = 45, which means that you want 100 = 1*45 + 9*(sum of digits in tens place) but that requires 55/9 = sum of digits in tens place, and 55/9 is not an integer. Does all of that make sense? If you have any questions about this or need more help, please write back and show me what you have been able to do, and I will try to offer further suggestions. - Doctor Vogler, The Math Forum http://mathforum.org/dr.math/
Date: 06/10/2004 at 11:03:30 From: Doctor Douglas Subject: Re: Digits 1-9 totalling 100 Hi Claire. Dr. Vogler's answer explains why it is impossible to obtain 100 if you are restricted to using only whole numbers. However, there is a trick that allows you to get a solution; you need to use a decimal point and a vinculum to get a repeating decimal. Check out these pages: Recurring Decimal Notation http://mathforum.org/library/drmath/view/62850.html 0.9999... = 1 http://mathforum.org/dr.math/faq/faq.0.9999.html Once you understand the notation, I don't think you'll have much difficulty finding a solution. - Doctor Douglas, The Math Forum http://mathforum.org/dr.math/
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