Latin boards, and the mathematician who loved them

A Latin board is the plainest puzzle in Quiltoku: no boxes, no extras, just rows and columns. It is also the oldest idea in the game, and it comes with one of the best stories in recreational mathematics.

The rule

On a Latin board every mark appears exactly once in every row and exactly once in every column. That is all. There are no thick outlines, because there are no boxes.

A 4×4 Latin board. The outlined square sits in the shaded row and column. The row already has green, blue and yellow. So does the column. Red is the only mark left.

Why it is harder than it looks

Take a regular puzzle and remove the boxes, and you have taken away a third of the clues' power. On a regular board, a square is checked three ways: its row, its column and its box. Here there are only two. Every given square rules out fewer possibilities, so the puzzle gets by on logic alone, with fewer shortcuts. Quiltoku makes up for it by handing out a few extra clues on Latin boards, but the feeling is different. Latin boards are open and airy. The deductions are long chains rather than quick corners.

The two techniques in the solving guide that matter most here are the first (what is missing from this row or column?) and the fourth (where can this mark still go in this row?). Boxes cannot help you, so the second and third techniques fall away.

A shape that fits any size

There is one thing a Latin board can do that a boxed board cannot. A boxed board needs a size that splits into rectangles: 4 into 2×2, 6 into 2×3, 9 into 3×3. A size like 5 or 7 cannot be split that way, because those numbers are prime. A Latin board has no such limit. That is why the Sunday daily is sometimes a 7×7: it exists only because the Latin rule needs no boxes.

Where the name comes from

The Swiss mathematician Leonhard Euler wrote about these squares in the eighteenth century. He used Latin letters to fill them, and the name stuck. A Latin square of order n is a grid of n rows and n columns with n different symbols, each appearing once in every row and once in every column. Sudoku is a Latin square with an extra rule added: the boxes. Every sudoku solution, in other words, is a Latin square. A Quiltoku Latin board is the same thing with the extra rule taken away.

The numbers get large quickly. There are 576 different Latin squares of size 4×4 and 161,280 of size 5×5. By 9×9 the count of ordinary sudoku grids alone is in the billions of billions, about 6.7 sextillion. You will never run out of boards.

Euler's thirty-six officers

In 1782 Euler published a puzzle that had been bothering him. Imagine thirty-six officers, six ranks from six regiments, one officer for each rank and regiment pair. Can they stand in a six-by-six square so that every row and every column has one of each rank and one of each regiment?

The trick is to see it as two Latin squares at once. One square holds the ranks and the other holds the regiments, and the two have to fit together so that every possible pairing of a rank and a regiment appears exactly once. Mathematicians call such a pair of squares orthogonal, and the combined grid a Graeco-Latin square, because Euler used Latin letters for one set and Greek letters for the other.

Euler knew how to build them for every odd size and for every size that divides evenly by four. He could not find one for sizes 6, 10, 14, 18 and so on, the sizes that leave a remainder of two when divided by four. He believed none existed, and he said so.

Euler was right once, and wrong all the other times

For 6×6 he was right. Around 1900, the French mathematician Gaston Tarry checked every possible arrangement by hand and showed that the thirty-six officers cannot be placed. It took enormous patience.

For everything bigger he was wrong. In 1959 and 1960 three mathematicians, R. C. Bose, S. S. Shrikhande and E. T. Parker, showed that the sizes Euler doubted do have solutions, all of them above 6. Parker found one of size 10 with the help of a computer. They became known as the "Euler spoilers". Today we know that Graeco-Latin squares exist for every size except 2 and 6.

A 10×10 square hangs in a quilt. The grid of the quilt that inspired this game is one of those "impossible" sizes: ten colours, two layers, every colour once in every row and column of each layer, and no pairing of a big square's colour with a small square's colour repeated. Read more on the logo page.

Playing with it

Latin boards first appear at Level 121, on a small board, and then come back in bigger and trickier forms. Some daily puzzles are Latin too. If you want a feel for the structure, try a Latin board and notice how quickly the whole grid is determined once a few rows are filled. You are doing, in small, the same thing Euler did on paper.