Mar 30,2006: UR "unique triple," 5-cell XY ring

[Ron Moore]

While plodding through the archives I found an interesting position in the March 30, 2006 Nightmare. Although I didn't come across the exact position below, which arises in Sudocue's solution, I'll use it since it illustrates the same points. One point of interest is that it provides an example which is lacking in the Solving Guide -- a "unique subset" pattern in which surplus digits on one side of a potential non-unique rectangle combine with two other cells to form, in effect, a naked triple in some house.

Code: Select all

.---------------------.----------------.------------------.
| 23459  *4569  &2369 | 7    *45   26  | 29    8     1    |
| 128    #68    &12678| 3     16   9   | 4     5     27   |
| 12459  *459    1279 | 12   *45   8   | 6     3     279  |
:---------------------+----------------+------------------:
| 7       2      36   | 456   8    13  | 15    9     456  |
| 34589  -45689  3689 | 456   2    13  | 1578  1467  4568 |
| 458     1      68   | 456   9    7   | 58    2     3    |
:---------------------+----------------+------------------:
| 129     7      4    | 8     36   26  | 12359 16    2569 |
| 128     3      5    | 9     167  4   | 1278  167   268  |
| 6      #89     1289 | 12    137  5   | 23789 47    2489 |
'---------------------'----------------'------------------'

In this case, the potential deadly pattern of "45" pairs lies in r13c25 (marked with "*" in the diagram); only one side of the rectangle contains surplus digits, 6 and 9; and cells r29c2 (marked with "#" in the diagram) combine with these surplus digits to produce a naked triple of {6,8,9} in column 2. These digits can be removed from r5c2. One way to view this is to observe that r1239c2 form an Almost Locked Set (ALS) of the five digits 4,5,6,8,9 in four cells. Digits 4 and 5 cannot simultaneously appear in this set as they would have to appear in r13c2 in some order, and this would imply a non-unique solution. Thus every other digit, namely, 6, 8, and 9 must appear in the ALS, so they can be eliminated from other cells in column 2 outside of the ALS. There is a small bonus -- since the two "6's" in the ALS are in box 1, the other "6" candidates in box 1 can be eliminated (r12c3, prefixed with "&"). Of course, this follows immediately after the first elimination of 6 from r5c2, since that leaves the box 4 candidates for "6" locked in column 3, but this need not be the situation in general.

Another point of interest in the position (reproduced below for convenience) is that it contains a 5-cell XY ring in r4c367, r6c37 (marked with "*" below). XY rings sometimes yield a large number of eliminations. For any side of an XY ring, the common value shared by the two vertex cells can be eliminated from any cell which sees both vertices. In this case, three of the sides produce one or more eliminations. The eliminations are marked with the "-" symbol placed before the eliminated digit.

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.-----------------------.---------------.----------------------.
| 23459  4569    23-69  | 7    45   26  |  29      8     1     |
| 128    68      12-678 | 3    16   9   |  4       5     27    |
| 12459  459     1279   | 12   45   8   |  6       3     279   |
:-----------------------+---------------+----------------------:
| 7      2      *36     | 456  8   *13  | *15      9     4-56  |
| 34589  45-689  3-689  | 456  2    13  |  1-578   1467  4-568 |
| 45-8   1      *68     | 456  9    7   | *58      2     3     |
:-----------------------+---------------+----------------------:
| 129    7       4      | 8    36   26  |  123-59  16    2569  |
| 128    3       5      | 9    167  4   |  1278    167   268   |
| 6      89      1289   | 12   137  5   |  23789   47    2489  |
'-----------------------'---------------'----------------------'