Arrange 1 to n² in a magic square and pour rain on it: the low cells hold water. Your browser searches for squares that hold more, saves your best ones to the record book, and can start from anyone's square.
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. Now think of each number as a column of stone that tall, and let it rain. Water runs off the outer edge, but a low cell walled in by taller ones holds water up to the lowest gap in its wall. Which magic square holds the most?
Craig Knecht brought the question to Al Zimmermann, who ran it as a contest, Retaining Water, from March to June 2010. 142 players took part. Walter Trump won, Hermann Jurksch came second, and as at many of Al Zimmermann's Programming Contests, the first prize was a sculpture by Bathsheba Grossman (see Lattice Foundry). The best scores from that contest still stand, and every one of them is on our record book, re-scored square by square.
It uses simulated annealing: make a small change, keep it if the square holds more water, and sometimes keep it even if it holds less, so the search can climb out of dead ends. It takes worse squares less and less often as it "cools". Small squares (up to 11 × 11) start from noise and are pulled into shape by a penalty for every line that doesn't add up. Bigger ones start as a magic square and only ever make swaps that keep every line's total the same.
Drag the square to turn it any way you like; scroll or pinch to zoom. Record book loads the record square for any size, or your own. Settings sets which sizes it tries and how long it searches. Sign in and your best squares are saved to the record book as you find them. A search that starts from somebody else's square is never saved, so nobody's name ends up on another player's work.
