###### Bifurcation

Bifurcation means “splitting into two”, which neatly describes your absolute last resort, only to be tried when none of the techniques you know can give you even the tiniest bit more of information about a puzzle.

When there’s nothing more you can deduce about a puzzle, you have to make an assumption about the contents of one of the squares, and proceed as if you know you’re correct. If you can solve the puzzle, well and good. If you can’t solve the puzzle with this assumption, go back to the point where you made the guess, make a different guess, and try again. Naturally, you’ll want to copy out the puzzle onto a fresh sheet of paper each time you make a guess, or you’ll end up with an absolute mess of scribbled and crossed out numbers. Let’s look at a diagram to make things clearer:

We can’t go can’t go any further with our puzzle, so we pick an unsolved square at random, in this case square [9, 8]. (It’s a lot easier if we pick a square which has only 2 possibles in it. We *could* pick a square which could hold any of six different numbers, but we’d only be making trouble for ourselves.) We don’t know whether it contains a 1 or a 3, so we make a blind guess that it’s a 1, and try to solve the puzzle. If that works, fine. If not, we try again, and this time we pretend that it’s a 3. This is a pretty tedious process, and in the worst possible situation, you might even find a puzzle where you had to bifurcate a couple of times in order to solve it! If you find such a puzzle, my advice is to give up – but they can be solved if you’re willing to stick with them.

Incidentally, this is the sort of thing that computer Sudoku solvers are good at. When programming a Sudoku solver, getting the computer to have a perfect memory and to be able to try many different combinations and backtrack when it gets stuck is trivial. It’s getting the computer to spot patterns and apply a bit of logic which requires the real work. For humans, it’s the opposite – most of us have a fair bit of intelligence and some pattern recognition skills, but keeping track of stacks of numbers and positions is tiring at best, impossible at worst.