Four Squares, One Lock
Four 3×3 grids, each a Latin square on 1, 2 and 3. Two carry a sum clue, two a product clue, and four small letters tie cells together across the squares. Solve all four, read the center numbers in order, and the lock opens. You get three tries, and the lock keeps count.
Take a square grid and fill it with symbols so that every row and every column contains each symbol exactly once. That is a Latin square, and the name is Leonhard Euler’s doing: in 1782 he filled his grids with Latin letters while chasing a parade-ground puzzle. Thirty-six officers, six regiments, six ranks — arrange them in a 6 × 6 formation so that no rank and no regiment repeats in any row or column. Euler decided it could not be done, and then guessed, more boldly, that the same was true for every square of order 6, 10, 14, and so on. He was right about 6 and wrong about everything after it, and the “wrong” took 177 years to establish. In 1959 three mathematicians — Bose, Shrikhande and Parker, nicknamed “Euler’s spoilers” — produced an order-10 counterexample, Parker’s found by about an hour of computer search. Scientific American put it on the cover and Martin Gardner wrote it up in his column, which is as close as a Latin square has ever come to being famous.
The squares turned out to be useful as well as decorative. In the 1920s the statistician R. A. Fisher laid out crop trials as Latin squares — each fertilizer once in every row and every column of the field — so that a damp corner or a sunny edge could not disguise itself as a difference between treatments. The same arrangement schedules round-robin tournaments and exam timetables, and it is the skeleton under Sudoku: a 9 × 9 Latin square with nine boxes bolted on. KenKen, invented by the Japanese teacher Tetsuya Miyamoto in 2004, drops the boxes and clues the square with arithmetic instead, which is where this week’s puzzle borrows its manners.
How many Latin squares are there? For a 3 × 3 grid on the numbers 1, 2 and 3, exactly twelve — small enough to list on an index card, and every one of them is a set of diagonal stripes. Twelve possibilities per square sounds like nothing. But chain four such squares together, give each an arithmetic clue, and insist that four pairs of cells agree across the squares, and twelve becomes 124 = 20,736 candidate boards with exactly one survivor.
This week’s puzzle is interactive. Four 3 × 3 squares: two clued by a sum, two by a product, and four pairs of cells tied together by matching letters. Fill all four, read the center numbers in order, and the lock opens. You get three tries per round.
No login, no timer. The checker grades the rules rather than looking up an answer, so a wrong try tells you only that something is still broken, not where. Pencil and paper work just as well, and the page prints cleanly.