Retaining Water.

The rules

• A magic square holds the numbers 1 to n², each once, so that every row, every column and both main diagonals add up to the same total.
• Each cell is a column of stone as tall as its number. Water poured on top runs off the outer edge, but a low cell walled in by taller ones holds water up to the lowest point of the wall around it. Neighbouring low cells can pool together and hold more than they would alone.
• Cells touch their 4 neighbours: up, down, left and right. Water doesn't leak out diagonally. Cells on the outer edge hold nothing.
• Your score for n is the total units of water the square holds.
• Send the numbers row by row. Al's 2010 format works as it is: (16,3,2,13), (5,10,11,8), (9,6,7,12), (4,15,14,1).
• For each n, only your best square counts. The contest ended in 2010, so there's no running score: it's all about the records.

Where this came from. Craig Knecht brought the problem to Al Zimmermann, who ran it as a contest from March to June 2010. Walter Trump won and Hermann Jurksch came second, and as at many of Al's contests, first prize was a Bathsheba Grossman sculpture. The records come from Al's final report, which kept every best square, so every record here has been re-scored by our scorer. Want a computer to search for you? The simulator in the Lab anneals squares in your browser, saves its best ones here, and can start from any square on this page.

The scoring code

As on the original site, here is the code that scores every entry, so you can check your own before you send it. It's written to be easy to read, not fast. Each version reads one entry per line and prints its score. We've checked all four against the site on all 102 squares in Al's final report, plus 210 altered ones, good and bad.

Why QBasic? It's the first programming language Neil, who runs ISS, ever learned, and this version is a nod to that. QBasic can't read typed-in input the way the others do, so it reads your entries from SQUARES.TXT and saves its answers in RESULTS.TXT.