When I went over to see a rare coin collection, I asked the care taker of the coins about them.
"What can you tell me about your coins?"
"In each box, the number of coins is a perfect square, and each box has a different number of coins."
"How are the coins in your 9 boxes organized?"
"I put them in chronological order by groups. The ancient coins are in boxes A, B, C; my old coins in boxes D, E, F; and my recent coins sit in boxes G, H, I."
"How else are they organized?"
"Well, within each time period (ancient, old, recent), the numbers form an arithmetic sequence, and the common difference is the same for all three time periods."
"How many coins do you have in each box?"
"I'm not sure, but I do know that a newer box would always have more coins that an older box from that time period. For example, feel box A. I have very few coins in that box, less than a dozen."
How many coins are in each box?
The posted solution is ingenious and does meet all the conditions of the problem as stated. However, the solution isn't within the bounds of reality, with over 16,000 coins in a single box. That box is a crate. :-)
Just a thought.