Coins in a Square Array

Date: 7/8/96 at 9:6:16
From: Anonymous
Subject: Prove that the butler lied

Dear Dr. Math,

A man says to his butler, "I left some valuable coins on the table 
this morning in a square array and now there are only two left."  The 
butler answers, "Three burglars came in, divided the coins equally, 
and left these two because they couldn't split them between them."  

I have to prove that the butler lied.

I know that I can prove it by showing that square numbers when 
divided by 3 do not ever have a remainder of 2, but I don't know how 
to write the formula. I also know that every third square number 
(multiples of 3 will be divisible by 3) won't have a remainder, but I 
don't know how to prove it mathematically.  Can you help?

Date: 7/8/96 at 16:56:5
From: Doctor Pete
Subject: Re: Prove that the butler lied

Clearly, every number leaves a remainder of 0, 1, or 2 when divided by 
3. Suppose n is divisible by 3.  Then n^2 is also divisible by 3.  
Note n+1 leaves a remainder of 1, and (n+1)^2 = n^2 + 2n + 1 will also 
leave a remainder of 1, since both n^2 and 2n are divisible by 3.  
Finally, n+2 leaves a remainder of 2, but (n+2)^2 = n^2 + 4n + 4 
leaves a remainder of 1 again, because 4 leaves a remainder of 1 when 
divided by 3.  Therefore every square leaves a remainder of 0 or 1 
when divided by 3, and never a remainder of 2.  Therefore the butler 
lied.

-Doctor Pete,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/