Locked Pair, Trio, Quartet, . . .

Terminology

Pair — when one Cell contains exactly two candies, those two candies are called a Pair.

Trio — when one Cell contains exactly three candies, those three candies are called a Trio.

Quartet — when one Cell contains exactly four candies, those four candies are called a Quartet.

Quintet — when one Cell contains exactly five candies, those five candies are called a Quintet.

(Similarly for Sextet, Septet, Octet, Nonet.)

Set — generic term for a Pair, a Trio, a Quartet, . . .

 

Locked Pair

External synonyms:  naked locked pair, naked pair.

A "Locked Pair" is not a Pair(!).  It is a set of two Cells in one House which contain identical Pairs.

If two Cells in one House both contain precisely the Pair {2 5}, then the digits 2 and 5 must be used up to fill those two Cells — so you can kill all the 2s and 5s occurring in all the other Cells of the House(!).

The two Cells containing {2 5} are called a Locked Pair.

Examples:

Locked Pair:  two examples. The Upper Left Block has a 2,5 Locked Pair.  One of the green Cells will have to hold the Block's final 2 BigNum, and the other green Cell will hold its 5; so you can kill all the other 2s and 5s in the Block. The fifth Row has a 1,7 Locked Pair; you can kill all the other 1s and 7s in the Row.

 

Locked Trio

A "Locked Trio" is not a Trio. It is a set of 3 Cells in one House whose candies (taken together) involve only 3 digits a, b, c.

The description of a Locked Trio is more subtle than that of a Locked Pair.

For starters, suppose a House contains three Cells each holding precisely the Trio {1 2 7}:  then obviously the three digits 1,2,7 will be used up to fill these three Cells — and so you can kill all the 1s, 2s, 7s occurring in the other Cells of the House. The digits 1,2,7 are "locked" in the original three Cells.

But then, suppose instead that the House contained three Cells holding precisely {1 2}, {1 2 7}, and {2 7}:  this is still a Locked Trio, because the digits 1,2,7 will still be used up to fill those three Cells(!). It's very important to digest this idea, and it does take some time to get used to it.

Examples:

Locked Trio:  two examples. The Middle Left Block has a 2,3,6 Locked Trio; the Block's 2, 3, & 6 BigNums are going to wind up in the green Cells, so you can kill all the other 2s, 3s, and 6s in the Block. The sixth Column has a 1,7,8 Locked Trio; you can kill all the other 1s, 7s, and 8s in the Column.

 

Locked Quartet

A "Locked Quartet" is not a Quartet. It is a set of 4 Cells in one House whose candies (taken together) involve only 4 digits a, b, c, d.

If you grasped the idea of a Locked Trio, then it will be clear what a Locked Quartet is. For example, four Cells in a House containing {2 5}, {2 7}, {5 7 9}, and {5 9} constitute a Locked Quartet in the digits 2, 5, 7, 9.

Examples:

Locked Quartet:  two examples. The second Row has a 1,2,7,8 Locked Quartet; you can kill all the other 1s, 2s, 7s, and 8s in the Row. The Middle Left Block has a 1,2,3,4 Locked Quartet; you can kill all the other 1s, 2s, 3s, and 4s in the Block.

 

One Locked Set in Two Houses

It's possible for a Locked Pair or a Locked Trio to fall within the bounds of two Houses, in which case it applies to both of them.

This situation occurs frequently.

Example:

One Locked Set in two Houses. The second Row contains a 2,7 Locked Pair; you can kill all the other 2s and 7s in the Row. But of course the Upper Left Block also contains the same 2,7 Locked Pair, so you can kill all the other 2s and 7s in the Block too.

 

Decomposable and Nondecomposable Locked Sets

If you think about the above remarks on Locked Pairs, Trios, and Quartets, you can see there's a general definition:

• Locked Set — N empty Cells of one House which contain only candies chosen from a specific set of N different candies.

This is a good general definition — it's useful in exactly that form for certain kinds of theoretical reasoning about Sudokus.

This definition has the interesting implication that the empty Cells of any House form a Locked Set (because the House is missing N BigNums, so these are the N candie values scattered throughout its N empty Cells.

But there is an important subtlety, the issue of decomposability:

• decomposable — A Locked Set is decomposable if it can be broken down into several smaller Locked Sets.

• nondecomposable — A Locked Set is nondecomposable if it cannot be broken down into smaller Locked Sets.

For example:

Decomposable and nondecomposable Locked Sets. Row 2 contains a 1,2,3,4,5,7,9 Locked Septet which can be decomposed into two Locked Pairs and a Locked Trio. Row 5 contains a 1,2,3,5,6,7,8 Locked Septet which can be decomposed into a Locked Trio and a Locked Quartet. Row 8 contains a 2,4,5,6,7,8,9 Locked Septet which is nondecomposable.

If you've been working Sudokus, then you will object that the above Grid is naïve because it doesn't show how the smaller Locked Sets actually arise (usually). So let's admit that. The following Grid is a simple example of how those various Locked Sets might have arisen:

Decomposed Locked Sets:  how they arise. (Precursor of the preceding Grid.) The red candies are there, but they're the ones that will be eliminated by the Locked Set reasoning. (LS = Locked Set.) Row 2:  the 2,5 LS causes the 1,4,9 LS to spring into existence, which then causes the 3,7 LS to spring into existence. Row 5:  the 3,6,8 LS causes the 1,2,5,7 LS to spring into existence. Row 8:  there are no smaller Locked Sets in this Locked Septet.

So you can see that decomposability is not a difficult concept; as the last Grid shows, Locked Sets in decomposed form are what arises in the course of normal Sudoku play.

 

Connection to Locked Single

On the page where a Locked Single is defined, I mentioned that the name of that Tactic would make more sense once a Locked Set had been defined (see preceding section).

A Locked Single is the world's simplest Locked Set:  it is one Cell that contains only one candie.

 

Naked Locked Set

Sometimes people call a Locked Set a Naked Locked Set, when they want to emphasize the contrast with a Hidden Locked Set. So for example if you see the term Naked Locked Pair, that's the same thing as a Locked Pair.

 

OK, So What Do We Do with All This?

When you start, the first thing you want to do is exhaust the Local Tactics before going on to the other Tactics. So do you do Locked Sets first, or do you do Claims first?

Rule of thumb:  when the candies in the empty Cells are not dense (meaning there are a reasonable number of Pair Cells and not very many Cells containing four or more candies), do Locked Sets first. This will clean up the Grid and make the search for Claims much easier.

Before applying Locked Sets really carefully, it's reasonable (and very satisfying) to just look over the Grid at random and apply every Hidden Locked Single you see (because it gives an immediate Crowning) and also apply any Locked Pairs or Trios you see, since they're so easy to spot.

But when you've picked off all the easy stuff, you should apply Locked Sets methodically:

• In every Block, crown every Hidden Locked Single, and then find and apply every Locked Set you see.

• Do the same in every Row.

• Do the same in every Column.

If any House has a lot of empty Cells and you didn't succeed in decomposing its big Locked Set, then it's worth looking for Hidden Locked Pairs in that House (see the page on Hidden Locked Sets).

That's 27 Houses you're processing. As you go, the results diminish in number, but there's often an unexpected Locked Set near the end, just when you think there aren't any more. It's worth it to persevere.