Number and Its Square Using All 9 Digits Exactly Once
Date: 05/22/98 at 05:34:11 From: Faye Subject: numbers Dr. Math, I want to know the way to get the answer for the following: Find all the whole numbers for which the number and its square together consist of exactly the nine digits 1, 2, 3,..., 9 appearing exactly once. Zero does not appear. From, Faye
Date: 05/22/98 at 13:40:16 From: Doctor Schwa Subject: Re: number and its square using all 9 digits exactly once Interesting question, Faye! I started by thinking about how many digits the original number would have. A three-digit number, since it's between 100 and 999, when squared is between 10000 and, well, a bit less than 1000000. So in order to have the right number of digits, we have to have a three- digit number whose square has six digits. Right away, we've narrowed down the number to only a few hundred possible possibilities, from 317 (the first one with a six-digit square) to 987 (the last one that doesn't repeat any digits). So then what? We could just painstakingly check each of those few hundred numbers, but that would get boring pretty fast. How else can we narrow the search? Well, I had one other idea. The last digit of a number is related to the last digit of a square, as 0^2 = 0, 1^2 = 1, 2^2 = 4, and so on. Among other things, that means that the number can't end in 0, 1, 5, or 6, because then the square would end in the same digit, but we can only use each digit once. That's about all the ideas I have so far ... it should give you a good start on doing the rest. If you're patient, you could just check all the remaining possible numbers on a calculator. Or, you could try cutting down the search some more (for instance, numbers between 317 and 444 can't end in a 1, because the square will start with 1). And they can't end in 9, either, because then the square will end with 1. So now let's see what we need to check in the 300s: 324, 327, 328, 342, 347, 348, 352, 354, 357, 358, 362, 364, 367, 368, 372, 374, 378, 382, 384, 387, 392, 394, 397, 398 Still rather a lot, especially considering that there are still the 400s, 500s, 600s, 700s, 800s, and 900s left to check. You can cross off a few more using the techniques I've already mentioned (348, for instance, since its square will end with 4, which was already used). Then apply a generous dose of patience with a calculator. If you hear of any better shortcuts for cutting down on the search, let me know! -Doctor Schwa, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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