# Latin square puzzles

A Latin square puzzle is any puzzle where some finite set of \(n\) numbers appear in an \(n \times n\) grid, such that each number appears exactly once per row, and once per column. Each puzzle type introduces one or more restraints on the Latin squares which are solutions. Variations on the Latin square theme include KenKen, Futoshiki, Cartouche, and likely many others that I've yet to hear about. Here are some strategies for solving Latin square puzzles, that apply to all variants:

Strategy 1:

Fill in the final value, for rows and columns missing only one number.

Strategy 2:

Look for a cell where most numbers appear along the row and column to which it belongs. For example, in a 3x3 grid, if there is a 2 along the row, and a 3 along the column, then the cell must contain 1. Sometimes there will be more than one possibility remaining. For example, if the same were true in a 4x4 grid, then both 1 and 4 would remain as possibilities.

Strategy 3:

Look for a cell where all possibilities for each cell along the row or column are accounted for. As an example, suppose you have a 3x3 grid. There is a 3+ region with two cells on the first row. We know the possibilities are (1, 2) and (2,1). But then a 3 can't go in either position, and hence, it must be placed in the remaining cell, which is outside the 3+ region.

Strategy 4:

If some row or column has n cells with exactly the same n possibilities, then none of those possibilities can be used in any other cell in that row. For example, suppose that three cells, in some row, have the candidates 2, 4, and 5. Then 2, 4, and 5 cannot appear anywhere else in the row.

Strategy 5:

Suppose a \(5 \times 5\) puzzle has a row where the remaining possibilities for each cell are

12345, 245, 1345, 24, 25

Notice that 1 and 3 occur in the 1st and 3rd cells only. Because we have 2 numbers occurring in exactly 2 cells, we can remove the other possibilities from these cells.

13, 245, 13, 24, 25

Let's reconsider the problem

12345, 245, 1345, 24, 25

Notice that 2, 4, and 5 occur in the 2nd, 4th, and 5th cells. Because we have 3 numbers occuring in 3 cells, where each cell has no more than 3 possibilities, we can remove these numbers from all other cells.

13, 245, 13, 24, 25

This suggests that the removal process is confluent. That is, the order in which you remove possibilities doesn't affect the final outcome.