So far we have been using one unit as target to do pointing out and/or fitting in, now we are going to add one dimension to it, a two-unit target!
See the board below, you can see both column 6 and row 7 are highlighted. Three empty cells and two missing numbers, 4 and 6, in the two-unit target. It seems locked: if 4 goes to 71 then 83 is 6 and 59 is 4, vice versa if you switch number 4 and 6. This situation often happen close to the end of the puzzle, make you attempt to guess.
See the board below, you can see both column 6 and row 7 are highlighted. Three empty cells and two missing numbers, 4 and 6, in the two-unit target. It seems locked: if 4 goes to 71 then 83 is 6 and 59 is 4, vice versa if you switch number 4 and 6. This situation often happen close to the end of the puzzle, make you attempt to guess.
No need of guess work in this case, the key of the lock is in box 4. There are two empty cells and two missing numbers, 6 and 8, in box 4. Number 8 won't help in this case but for number 6 we have 2 assumptions:
- If 6 goes to 44, then 71 cannot be 6, then 83 must be 6
- If 6 goes to 49, then 59 cannot be 6, then 83 must be 6
Both assumptions lead to the same solution: 836. Click the board to see it in animations.
If you could click the board again and look at the two-unit target, one of the three empty cells 83 is sharing between column 6 and row 7, the other two empty cells 71 and 59 are stretching out like 2 wings. I call it Angel's Wings. When locate Angel's Wings, we need a pointer that can point out both of the wings at the same time box 4 is just what we need, with its 44 point out 71 and 49 point out 59. I call it Double Pointer.
When you find Double Pointers that can point out both of the Angel's Wings, you find the angel!
- If 6 goes to 44, then 71 cannot be 6, then 83 must be 6
- If 6 goes to 49, then 59 cannot be 6, then 83 must be 6
Both assumptions lead to the same solution: 836. Click the board to see it in animations.
If you could click the board again and look at the two-unit target, one of the three empty cells 83 is sharing between column 6 and row 7, the other two empty cells 71 and 59 are stretching out like 2 wings. I call it Angel's Wings. When locate Angel's Wings, we need a pointer that can point out both of the wings at the same time box 4 is just what we need, with its 44 point out 71 and 49 point out 59. I call it Double Pointer.
When you find Double Pointers that can point out both of the Angel's Wings, you find the angel!
In the board below, the angel hides in column 2 and row 6, but it needs two cell pointer 16 and 81 to reveal the Angel's Wings, head in 48, wings in 68 and 75, click the board to see it. Box 9 is the Double Pointer that points out both of the wings 68 and 75, we find the angel 6 in 48.
Box 9 also shows you the beauty of Double Pointer: it does not matter how many empty cells in the pointer unit, as long as all the empty cells can form two beams pointing to two directions.
Another example, this time the angel hides in box 2 and row 1, it takes fitter 1 and 9 in column 5 to reveal the Angel's Wings. Row 4 does the double pointing job and we find the angel 4 in 21.
Revealing Angel's Wings can be complicated, click the board below to see how it's revealed in column 4 and row 1, and click the board again to see how column 8 does its double pointing.
Finding the Double Pointer is not always easy either, click the board below to see Angel's Wings in column 1 and row 1, and click the board again to see how box 5 comes to be the Double Pointer.
And Double Pointers, relay, click the board below to see Angel's Wings in column 4 and row 7, and click the board again to see how Double Pointer row 5 points out one wing 83, relays to box 9, which relays to column 2, which points out the other wing 27, finally we find angel 6 in 81.
That's the Angel's Wings with Double Pointers, a brand new technique, but think about it, it's still Pointing Out technique, extended into two-dimension!
Technique No.1 Pointing Out - Easy Level
Technique No.2 Pointing Out - Medium Level
Technique No.3 Fitting In - Medium Level
Technique No.4 Fitting In - Hard Level
Technique No.5 Pointing Out - Hard Level
Technique No.6 Pointing Out - Expert Level
Technique No.7 Fitting In - Expert Level
The puzzles in this post are from http://school.maths.uwa.edu.au/~gordon/sudokumin.php The work is licensed under a Creative Commons Attribution 2.5 License. Attribution goes to Gordon Royle and The University of Western Australia.
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