Nine Digits, Perfect Squares -- and Better Strategies

Date: 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/