Smart Fork

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The discussion of the Smart Fork Tactic on this page assumes that you know what a Twin is and that you know how to do Twin Tagging. Read the page on Twin Tagging before going through the material below.

 

Twin-link Degree

A candie in any Cell can have at most 3 distinct Twins:  a Row Twin, a Column Twin, and an obliquely located Block Twin. Of course, it could have no Twins at all. A candie could have 0, 1, 2, or 3 Twins, and that number is the candie's Twin-link Degree.  Sometimes we will say that a candie is singly, doubly, or triply Twin-linked.

An empty Cell's Twin-link Degree is the ensemble of the Twin-link Degrees of each of its candies.

We will only need to think about the Twin-link Degree of Pair Cells:

There is no distinction between the Twin-link Degrees (p,q) and (q,p): a Twin-link Degree of (3,2) is the same as a Twin-link Degree of (2,3).

 

Fork

The general idea of a fork is this:

(To do this, of course, you need to have made two photocopies of the Grid.)

"See what happens":  what can happen?  One of three things:

So let's define a dumb fork:

What will a dumb fork get you? Usually, two Stalemates — nothing but a waste of time.

Or could you get lucky with a dumb fork? Yes, you could. But why depend on dumb luck when there's a better way?

 

Smart Fork

We're still interested in the possibility of doing a fork, but we'd like to do it in a way that makes sense. We want to do the fork at a Fault Point, where the Grid is weak and we have a good chance of cracking the whole Sudoku.

A Fault Point is a Pair Cell that has a high Twin-link Total.

The Twin-link Total of a Pair Cell is the sum of the Twin-link Degrees of its two candies:

This definition implies that a (1,3)-Degree cell and a (2,2)-Degree Cell are equally desirable (for the purposes of starting a fork).  Why?  Because in a Pair Cell when you crown one candie and kill the other, all the Twins of each of the two candies are determined. (Crown a candie, and all its Twins get killed; kill a candie, and all its Twins get crowned.)  So the Twin-link Total of a Pair Cell measures the number of external Twins directly reached when that Cell is used to start either branch of a fork.

So what's a Smart Fork?  It's a fork done at a Fault Point of Twin-link Degree (3,3) or (2,3). In this case, my actual experience says that you usually get the following results:

I said "usually":  that's not clear enough.  It's very rare for this Tactic to fail. After doing several dozen Smart Forks, I've encountered a Smart Fork failure for only one case of a Degree-(3,3) Fault Point and for three cases of a Degree-(2,3) Fault Point. What kind of failure?  Just this:  one of the two branches fails (produces a Stalemate).

And I said "produces" the Solution or a Conflict. That also needs clarification:  what kind of moves are involved in carrying out either branch of the Fork?  The good news is this:

The less good news is this:

Don't overlook the good news side of this:  when you have a hard Sudoku that you have not been able to crack with a Chain and it happens to have a (3,3)- or (2,3)-Degree Fault Point, extremely often a Smart Fork will yield a clear Solution and a clear Conflict where both are arrived at by easy moves.

What about lower-quality Fault Points? For a Fault Point with a Twin-link Total of only 4, you run a much higher risk that one branch of the Fork will yield a Stalemate. Nevertheless, I do give one example of a Twin-link-Total-4 case below.

So how do you do a Smart Fork?

Now we'll look at some actual cases.

 

Examples

When am I likely to resort to a Smart Fork?  When I've got a hard-to-solve Sudoku, I always try to find a Chain, or a succession of Chains, that will crack it. If I'm really unable to do that, then I'll try a Smart Fork if there exists a good Fault Point on the Grid.

The first example is an optimal case — it's a hard Grid, but it has a Fault Point with a Twin-link Degree of (3,3):

The yellow Cell in this Grid is an optimal Fault Point:  it has a Twin-link Degree of (3,3).

For this {5 6} Fault Point —

  • picking 5 yields the Solution;
  • picking 6 yields Conflicts.

I won't take up space by showing you the Solution Grid, because you would learn nothing at all by looking at it.

As for the Grid with Conflicts, I'll say this:  I had to nearly complete the Grid before reaching a Conflict — no wonder I had been unable to find a Chain to crack this Sudoku. And I won't display the Conflict Grid here either, because the places where Conflicts will finally show up depend on the order in which you choose to make moves.

A side remark on the notation in the Grid:  in addition to Twin-tagging the candies in the Fault Point {5 6}, I also partially Twin-tagged the external 5- and 6-Twins of the candies in the Fault Point just so you could see that those Twins were really there. But when you're checking the Grid for Fault Points, you should tag only the Fault Points themselves — this gives you an immediate view of how good each Fault Point is.

In the next example, the best Fault Point has a Twin-link Degree of only (2,3):

The best Fault Point in this Grid is the yellow Cell, which has a Twin-link Degree of (2,3).

For this {6 8} Fault Point —

  • picking 8 yields the Solution;
  • picking 6 yields Conflicts.

For this Grid, when I made the wrong choice R2C8 = 6, Conflicts developed in the Lower Center Block. So I decided to go back to the original Grid and try to construct a Wrong Chain starting from the Fault Point. I finally found one, running down the Right Tower, into the Lower Center Block, and back into the Right Tower. It wasn't simple — it had 11 Cells, and it involved the creation of three Locked Sets on the fly.

When the Chain you can find (for solving the Sudoku) is so long that it's as big as the area filled up to reach Conflicts in the Smart Fork Conflict Grid, one could certainly argue that the Smart Fork is just as aesthetically pleasing as the Chain is.

(See the Chain page for an explanation of Wrong Chains.)

In the third example, the three best Fault Points all have a Twin-link Degree of (2,2):

The best Fault Points in this Grid are the three yellow Cells: they all have a rather low Twin-link Degree of (2,2).

The two {3 6} Cells are equivalent — if you determine one, you have determined the other. So effectively there are only two independent Fault Points.

For the {5 7} Fault Point —

  • picking 7 yields the Solution;
  • picking 5 results in a Stalemate!

For the left (R6C5) {3 6} Fault Point —

  • picking 6 yields the Solution;
  • picking 3 yields Conflicts.

In this Grid, we were dealing with relatively low-quality Fault Points — they were only of Degree (2,2).  In a sense the {5 7} Fork turned out all right because in this particular case one branch yielded a Solution, but in general Fault Points with a low Twin-link Degree are not a safe bet for launching a Smart Fork.  A Fault Point of Degree (2,3) or (3,3) is vastly more likely to yield a clear result for each branch of the Fork.

That notwithstanding, what information did I get out of the Conflict Grid for the wrong choice R6C5 = 3? Well, I had to trapse all the way through the wrong-choice Grid until I finally developed a couple Conflicts in the last few Cells. Correspondingly, when I tried, in the original Grid, to construct a Wrong Chain to kill the 3 in R6C5, I had no luck whatsoever. So a Smart Fork was my only chance at cracking this Grid.

 

So What Does a Smart Fork Failure Look Like?

By failure, I mean that one of the two branches of the Fork produces a Stalemate.

The following case is fairly typical of the few Smart Fork failures I've seen with a Fault Point of Twin-link Degree (3,3) or (2,3).

This Grid has a high candie density (meaning not many Pair Cells and a significant number of Cells with four or more candies), but unfortunately this simple warning sign is not, in general, a guarantee of Smart-Fork failure.

The yellow Cell in this Grid is a Fault Point with a Twin-link Degree of (3,2).

The average candie density in the empty Cells is fairly high, which of course is not good.

For the {6 7} Fault Point —

  • picking 7 results in a Stalemate!
  • picking 6 yields Conflicts.

Sole result:  we know that the yellow Cell = 7.

Smart Forks usually proceed with easy moves, but I had to pull out all the stops here:

This is certainly not what you want from a Smart Fork. The only consolation is that it doesn't happen often.

In an effort to achieve closure, let's look at the Grid resulting from the right-choice Stalemate:

Preceding Grid modified by setting R2C2 = 7 and making all available non-Chain moves.

Now there is a new good Fault Point (the yellow Cell).

But this time it's possible to find a Gotcha Chain ABCDEFG to kill the 8 in the new Fault Point.

The Gotcha Chain has two heads, A and G, and if one of them isn't an 8, then the other is (if A=1, then follow the Chain ABCDEFG to see that G=8).  So the 8 in the yellow Cell is killed.

With the new Fault Point cracked, the entire Sudoku easily falls apart.

(As noted previously, you can read more about Gotcha Chains in the Chain page.)

So our Smart Fork clearly identified the wrong choice, but the right choice left us with an unsolved Grid that was still difficult. The only saving grace, such as it is, is that now we are able to find a Chain that finally cracks this Sudoku.

 

What Do We Make of All This?

If you accept a Smart Fork as a reasonable Tactic for solving a hard Sudoku which you're unable to crack with a Chain, then you're best off limiting yourself to Fault Points of Twin-link Degree (3,3) or (2,3), since these have a very high success rate. As noted above, Twin-tag only the actual Fault Points, so you'll see what your choices are after scanning the Grid.

But even if you reject the Smart Fork as a Tactic (because of the photocopies or for philosophical reasons), there's still a reason to try a Smart Fork a couple of times, not really to solve a Sudoku, but solely for thinking purposes. For a hard Grid, the wrong branch of a Smart Fork can require a surprisingly large number of moves to finally reach a Conflict:  this can produce a grudging admiration for the subtlety of this particular Grid and give you some inkling of why you couldn't find a Chain to crack it.

Beyond all that, there's this:  the basis of a Smart Fork is a Fault Point, which is a point where the Grid structure is so weak that cracking the Fault Point might result in cracking the whole Sudoku. But what that means is that if the Grid has a high-Degree Fault Point, we actually wouldn't try a Fork first, but rather we'd try to construct a Gotcha Chain to crack the Fault Point.  An example of this was shown above in the second Grid in the section So What Does a Smart Fork Failure Look Like?, and four more examples are discussed in the Chain page.

 

 

This page was last updated on 2011 January 7.

The home page for this site is   alcor.concordia.ca/~stk/sudoku/

 

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