# How Sudoku Works

## Solving a Sudoku: Diabolical and Beyond

Until recently, some of Michael Mepham's diabolical-rated sudokus in London's Daily Telegraph could not be solved by logic alone. They actually required guessing at a certain point, which sudoku purists consider a real no-no. Due to the amount of controversy (and hate mail) Mepham received, he has stopped publishing puzzles that require guessing. Still, the process of solving one of these puzzles is interesting, if only because you have to know enough to be absolutely sure there are no more clues before you begin the guessing process. Mepham called the strategy "Ariadne's Thread" (see below), which entails picking one of two possible solutions for a given square and following it until you reach a solution or a dead end. If you reach a dead end, you retrace your steps to the guessing point and pick a different number.

In Michael Mepham's "Book of Sudoku 3," there's a diabolical sudoku that starts like this:

Using logic, you can get here:

But there are no more logical clues -- we're stuck. The only option left to us is to guess -- and leave our numbers penciled in so we can follow Ariadne's Thread back to our starting point if our guess turns out to be wrong. If we choose one of the squares with only two options, we've got a 50/50 chance of picking the right number. Let's go to row 2, column 1 and pick the 4. Assuming that 4 is the solution to the 2,1 square, we can solve a bunch of other squares by extension -- but we end up with a problem.

If the solution at 1,7 is a 4, then the solution at 6,7 has to be the 5. But row 6 already has a 5. So now we need to erase the solutions we drew from our guess and go back to square 2,1. This time, we'll pick the 5.

The 5 is indeed the solution to the square at 2,1, and it allows us to solve the entire puzzle.

Although Mepham stopped publishing guess-necessary sudokus in his column, you can still access them at Mepham's sudoku Web site, sudoku.org.uk.

In response to the tremendous popularity of sudoku, different versions of the puzzle have emerged to provide even more of a challenge. One type of "extreme sudoku" is 3-D sudoku. Just line up nine complete sudoku grids into a three-dimensional cube that requires complete rows, columns and boxes on three interconnected axes, and you've got yourself a 3-D sudoku puzzle. All of the same rules apply, but now you're working on multiple planes. To solve the cube, you need to work out the solutions for each of the nine grids individually, although if you download one of the dozens of 3-D sudoku computer programs, you can work on the puzzle in full 3-D glory.