Nine Digits, Perfect Squares -- and Better StrategiesDate: 03/07/2010 at 20:57:57 From: aqsa Subject: trying to find out how to solve a puzzle What trio of three-digit perfect squares includes each of the digits 1 through 9 exactly once? I know 169 is a square, and 13 is its square root. I know 324 is a square, and 18 is its root. This leaves the digits 5, 7, and 8. I've tried to see if any combination of those three numbers will turn out to be a square, but I'm not sure what to do right now. I've tried guess-and-check, and it's taking a very long time, so I was wondering if you could show me a formula or something similar so I can solve this problem faster. Date: 03/07/2010 at 23:23:01 From: Doctor Peterson Subject: Re: trying to find out how to solve a puzzle I took it in the opposite direction: I made a list of all three-digit squares (that is, the squares of numbers from 10 through 31), and crossed off those that have repeat digits (like 100). That left 13 squares that might work. Then I looked at which of these share no digits with others in the list (like you found with 169 = 13^2 and 324 = 18^2). There are several ways you could keep track of these, but I drew lines between pairs of numbers that are "compatible" in this sense. Finally, I looked for groups of three numbers that are all compatible with one another. (At first I found none, but then I went through the list and discovered I'd left off one pair; and when I drew that one more line, I had a solution!) See if you can do something like that. You might even find a better way than mine. The benefit of my method is that I narrowed down the possibilities quickly, and I had an orderly way of finding ALL possible triples, rather than trying things randomly. Those are useful strategies for many problems -- as is the idea of turning a problem around and looking at it from a different perspective. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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