‘The dog did nothing in the night-time.’

‘That was the curious incident,’ remarked Sherlock Holmes.

-Conan Doyle, Silver Blaze

I have just spent several weeks attacking the eleven sudokus published by Mr. Mepham on the sudoku.org.uk website and identified as being “unsolvable” without bifurcation, with the hopes that interested players would devise non-bifurcative techiques to solve them. One of the techniques I found to attack the puzzles uses a peculiar property of closed loops of pairs of candidate cells for a number. If the number of cells in the loop is even - that is there are four, six, etc., in the loop - then it is quite possible for all the pairs in the loop to be conjugates - pairs of cells for the candidate that are the only possible locations for that number in the column, row or box they share. An X-wing, for example, is a closed loop of four cells linked in conjugate pairs, while the rarer "Swordfish" is a closed loop of six cells in conjugate pairings.

Strangely enough, this is not possible for a closed loop of five paired cells - sometimes referred to as a "Turbot Fish" - or for any longer loop of an odd number of cells. The reason is that if all the pairings were conjugate, there would be no way to place a number in any of the looped cells without generating an immediate contradiction that would make a valid solution impossible. And we can draw an important inference from what the loop doesn’t do: there must always be one or more other cells for the candidate number that prevent some pairing or pairings from being conjugate by sharing the same column, row or box jointly occupied by the pair. I refer to these as “guardian” cells. Now, logically, it can be concluded that at least one (maybe more) of these “guardians” cells must really contain the candidate number (or the loop would no longer be silent: it would in fact rip the sudoku to shreds).

That means that if there is only one guardian cell the candidate number can immediately be installed in that cell, and if more than one guardian cell is identified, the candidate number can be erased from any cells that are “seen” by all the guardians.

I call this technique the “Broken Wing” - both because the five-cell version looks like an X-wing with one arm twisted, and because the conclusion depends on the fact that the perfect conjugate loop must be broken at one or more places. Unlike colouring, which also loves pairings, the technique is non-bifurcative - you do not trace possible effects of possible placements along the loop, but rather you identify the loop’s existence, note the guardians and (hopefully) either immediately install a candidate or remove one or more. The shape is as easy to identify and use as the X-wing, and from its prevalence in Mr. Mepham’s “unsolvables” [three of them can be solved outright using this approach] I suspect it can be used to short-circuit a number of daily newspaper puzzles.

Here are a few examples of the technique in practice. [I apologize for the graphics. For a far better graphic presentation, I’d invite you to check the several threads on the “unsolvables” in the “Eureka” section of the discussions forum at sudoku.org.uk. All but the very first postings are supported by JPEG graphics]

The first is Mr. Mepham’s “unsolvable" number six at the point where standard techniques run out and a resort to bifurcation seems necessary. It is as “spectacular” a version of the "Broken Wing" technique in practice as there can be, since there is only one “guardian” cell preventing a perfect loop.

The five candidate cells for the number 7 forming a loop are marked with an X. There is only one “guardian” cell (marked with a G) preventing an impossibly-perfect conjugate pairing, at r2c1. Accordingly a 7 may immediately be installed there. As can be seen, there is no bifurcation in this approach, as we need not guess at all about the cells in the loop - we’re looking elsewhere. And even there we’re not guessing - we’re looking for one-step inevitabilities.

17 X |..2..|..9..||...6...|.17X.|..5..||..3..|..8..|..4

178G|..3..|.78.||..17..|...9...|..4..||..6..|..2..|..5

...5...|..6....4..||...8...|...3...|..2..||..7..|..1..|..9

- - - - - - - - - - - - - - - - - - - - - - - - - - - - -

...3...|..4..|..5..||...9...|...6...|..8..||..2..|..7..|..1

...6...|..7..|..2..||...3...|...4...|..1..||..9..|..5..|..8

...9...|..8..|..1..||...2...|...5...|..7..||..4..|..3..|..6

..48..|..5..|..6..||...4...|...2...|..3..||.18.|..9..|..7

- - - - - - - - - - - - - - - - - - - - - - - - - - - - -

47X..|..1..|..3..||.47X.|...8...|..9..||..5..|..6..|..2

...2...|..9..|.78.||...5...|.17X.|..6..||.18.|..4..|..3

A second example is perhaps more typical of the “Broken Wing” reasoning process is in Mr. Mepham’s “unsolvable” number three, below. The pattern of five candidate cells, coincidentally again for the number 7, is once more marked with X’s. This time there are two “guardian” cells at r2c1 and r9c9. We don’t know which of the two is “real” it could be that both are, but since both of them “see” the cells at r2c9, that cell may be eliminated as a 7, and that in turn means we can install a 7 at r1c7. That in itself doesn’t solve the puzzle, but it’s still a good illustration of the action of the “Broken Wing” technique and the ease with which it can be applied, at least for rings of 5 cells.

.279X.|...3...|..29..||..8..|..5..|..4..||.79X.|...1...|...6

.578G.|578X|...1...||..6..|..2..|..9..||...4...|...3...|578T

....4....|...6...|.579.||..3..|..7..|..1..||.589.|...59..|...2

- - -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

...56...|...4...|...7...||..2..|..9..|..3..||..56..|...8... |...1

25689|.589.|2589||.45.|..1..|..7..||...3...|4569|..59

...59...|...1...|...3...||.45.|..8..|..6..||...2...|4579|.579

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

....1....|.589.|.589.||..7..|..3..|..2..||5689|.569.|...4

...57...|...2...|...6...||..9..|..4..|..8..||...1...|..57..|...3

....3....|789X|...4...||..1..|..6..|..5..||789X|...2...|789G

The pattern is extremely distinctive for five cells, since it always involves four boxes in a rectangle, three of them containing a single cell of the loop and the fourth holding a non-aligned pair. Finding the guardians is a matter of inspection from there; whether they yield any useful information depends on their number and location. Usually, two or three of the pairings must be conjugate for the guardians to eliminate a candidate cell, but that seems to happen surprisingly often.

There are numerous other examples of this “Broken Wing” technique for five cells in the eleven “unsolvable” puzzles posed by Mr. Mepham, and at least two for seven cells. Judging by that sample, I’d suggest that it will turn up in many other puzzles if it is looked for, and may be particularly useful if you’re trying to avoid “colouring” or “forcing chains”.

You must excuse me now: I have to go listen for the dog.

Rod Hagglund