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David Bryant
Joined: 29 Jul 2005
Posts: 559
Location: Denver, Colorado
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 Posted: Fri Apr 07, 2006 6:49 pm Post subject: 7 Apr Nightmare -- an "anti-unique rectangle" |
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This puzzle is extremely tough, even by "nightmare" standards.
After making all the more or less "obvious" moves I arrived at this position.
| Code: | 2379 6 249 5 28 248 3789 1 389
259 1279 1259+ 3 1268 1268 589 579 4
35 14= 8 7 9 14~ 2 356 356
4 139 7 2 1568 1368 359 3569 3569
69 5 169- 146 7 136* 349 8 2
8 23 26 46 56 9 1 3456 7
259 279 3 189 4 12 6 579 1589
1 249 24569 689 236 7 34589 3459 3589
679 8 469 169 136 5 3479 2 139 |
Here there's a "fork" on the "1"s in row 3 and column 3 that lets us eliminate the possibility of a "1" at r5c6. Then we can use a double-implication chain from r6c4 to show that r5c6 = 3:
A. r6c4 = 6 ==> r5c6 <> 6.
B. r6c4 = 4 ==> triplet {1, 6, 9} in r5c1, 3 & 4 ==> r5c6 <> 6.
Now the grid looks like this.
| Code: | 2379 6 249 5 28 248 3789 1 389
259 1279 1259A 3 1268 1268 589 579 4
35 14 8 7 9 14B 2 356 356
4 139 7 2 1568 168 359 3569 3569
69 5 169a 146 7 3 49 8 2
8 23 26 46 56 9 1 3456 7
259 279 3 189b 4 12* 6 579 1589
1 249 24569 689 236 7 34589 3459 3589
679 8 469 169b 136 5 3479 2 139 |
Next we have a "Nishio" move. Notice that there are only two places for a "1" in column 3.
A. r2c3 = 1 ==> r3c6 = 1 ==> r6c6 <> 1.
B. r5c3 = 1 ==> r7c4 or r9c4 = 1 ==> r6c6 <> 1.
We conclude that r6c6 = 2; this allows us to eliminate some "2"s in the top left 3x3 box (because the "2" in column 1 is at r1c1 or r2c1), to identify a {5, 7, 9} triplet in row 7 (uncovering a hidden pair {1, 8}), and to eliminate the "5" at r2c8 via a "coloring" argument.
Now the puzzle looks like this.
| Code: | 2379 6 49 5 28 48 3789 1 389
259* 179* 159 3 1268 168 589 79 4
35 14 8 7 9 14 2 356 356
4 139 7 2 1568 168 359 3569 3569
69 5 169 146 7 3 49 8 2
8 23 26 46 56 9 1 3456 7
59* 79* 3 18 4 2 6 579 18
1 249 24569 689 36 7 34589 3459 3589
679 8 469 169 136 5 3479 2 139 |
Here there's a very unusual formation in r2c1&2, and r7c1&2. I think it should be called an "anti-unique rectangle." Observe that we cannot have r2c1 = 5 and r2c2 = 7, because that would force two "9"s in row 7.
This is where it gets very cute.
A. r1c1 = 2 ==> r1c5 = 8 ==> r1c6 = 4 ==> r1c3 = 9.
B. r1c1 = 2 & r1c3 = 9 ==> r2c1 = 5.
C. r1c1 <> 7 ==> r2c2 = 7.
So we can't have a "2" at r1c1, because of the "anti-unique rectangle." This means that the "2" in the top left 3x3 box must appear at r2c1, and that in turn forces a "2" at r1c5.
| Code: | 379 6 49 5 2 48 3789 1 389
2 179* 159 3 168 168 589 79 4
35* 14 8 7 9 14 2 356 356
4 139 7 2 1568 168 359 3569 3569
69 5 169 146 7 3 49 8 2
8 23 26 46 56 9 1 3456 7
59* 79* 3 18 4 2 6 579 18
1 249 24569 689 36 7 34589 3459 3589
679 8 469 169 136 5 3479 2 139 |
Now the "rectangle" is a bit lop-sided (a trapezoid), but it still works:
A. r1c1 = 3 ==> r3c1 = 5.
B. r1c1 <> 7 ==> r2c2 = 7.
So the "3" in column 1 can't appear at r1c1, and it must go in r3c1. This in turn implies that r2c3 = 5, and the rest of the puzzle is relatively straightforward -- for a little while! After making the obvious moves from here we arrive at one more very tough spot.
| Code: | 79 6 49 5 2 48 3789 1 389D
2 179 5 3 18 6 89 79 4
3 14 8 7 9 14 2 56 56
4 39A 7 2 18 18 359 356 3569
69 5 1 46 7 3 49B 8 2
8 23 26 46 5 9 1 34 7
5 79 3 18 4 2 6 79C 18
1 249 2469 89 36 7 34589 345 3589
679 8 469 19 36 5 3479 2 139 |
Another "Nishio" move -- on the "9"s -- finally breaks through the logjam.
A. r4c2 = 9 ==> r5c7 = 9.
B. r4c2 = 9 ==> r7c8 = 9.
C. r5c7 = 9 & r7c8 = 9 ==> r1c9 = 9.
But with "9" at A and D above there's no way left to place a "9" in the top left 3x3 box. So the "9" in the middle left 3x3 box must appear at r5c1.
Oh -- notice that the argument on the "9"s can be operated in the other direction, although it's a bit more complex that way. Start with the two spots for a "9" in row 7.
A. r7c2 = 9 ==> r4c2 <> 9.
B. r7c2 <> 9 ==> r7c8 = 9 ==> either r4c9 = 9 or else r1c9 = 9.
B1. r4c9 = 9 ==> r4c2 <> 9.
B2. r1c9 = 9 ==> r2c2 = 9 ==> r4c2 <> 9.
Has anyone noticed this sort of "anti-unique rectangle" before? dcb |
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Steve R
Joined: 24 Oct 2005
Posts: 289
Location: Birmingham, England
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| Posted: Sun Apr 09, 2006 4:31 pm Post subject: An "anti-unique rectangle" |
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I cannot remember seeing the patterns you mention and have been playing with Xs and Ys.
The second anti-unique rectangle (trapezoid) elimination is straightforward:
| Code: |
-----------------
| Zetc |
| Zetc | Only two places for Z in the box
| YT |
|-----------------|
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| XY XZ |
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T can be eliminated from the top left Zetc. Incidentally, the YT cell is not required to be in the box, just in the left hand column.
I have not made much progress with the first anti-unique rectangle: probably not seeing the wood for the trees.
However, the elimination of 1 from r7c6 caught my eye. As far as pattern merchants are concerned it seems to be an extension of the empty rectangle, that is a box in which a candidate X is confined to one row plus one column.
The basic empty rectangle format is
| Code: |
-------------------------
| . . . | . . 1 | . . . |
| . . 1 | . 1 . | . . . |
| . . . | . . 1 | . . . |
-------------------------
| . . . | . . . | . . . |
| . . 1 | . . x | . . . |
| . . . | . . . | . . . |
-------------------------
| . . . | . . . | . . . |
| . . . | . . . | . . . |
| . . . | . . . | . . . |
-------------------------
Only two cells for 1 in column 3
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This eliminates 1 from r6c6, which is of no use in the puzzle itself. I do not recall coming across an extension like
| Code: |
-------------------------
| . . . | . . 1 | . . . |
| . . 1 | . 1 . | . . . |
| . . . | . . 1 | . . . |
-------------------------
| . . . | . . . | . . . |
| . . 1 | 1 . . | . . . |
| . . . | . . . | . . . |
-------------------------
| . . . | . . . | . . . |
| . . . | 1 . x | . . . |
| . . . | . . . | . . . |
-------------------------
Only two cells for 1 in columns 3 and 4
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eliminating 1 from r8c6, still less an extension of the grouped variety
| Code: |
-------------------------
| . . . | . . 1 | . . . |
| . . 1 | . 1 . | . . . |
| . . . | . . 1 | . . . |
-------------------------
| . . . | . . . | . . . |
| . . 1 | 1 . . | . . . |
| . . . | . . . | . . . |
-------------------------
| . . . | 1 . x | . . . |
| . . . | 1 . x | . . . |
| . . . | 1 . x | . . . |
-------------------------
Only two cells for 1 in column 3 and only two places for 1(cell and box) in column 4
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that you actually used in the puzzle.
Nice work.
Steve |
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