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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
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Associated Topics:
High School Exponents
High School Number Theory
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