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Naked Double Loop |
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The Naked Double Loop is a loop of paired candidates where two cells at a time act as a node, and where the two values from each connection (four values total) are the only values in the cells. Then any candidate that is not part of the loop, but shares units with connections in the loop, can be eliminated.
This technique is complex to explain, but fairly to simple to spot, so we'll use an example. The above image is taken from the famous puzzle known as Easter Monster. The naked double loop is highlighted in yellow and the eliminations that come from it are highlighted in green.
Let's begin with the second column from the left. In this column, the two values tying the top-left box to the bottom-left box are 4 and 8. Now let's look at the two cells from this column in the top-left box. Besides 4 and 8, the only candidates in these cells are 2 and 7, which happen to tie us to the other two cells in this box. From those two cells, besides the 2 and 7, the only candidates they have are 3 and 8, which tie into the cells on the far-right of that row. This pattern continues around the board until it ties back into our original cells. Because each pair of cells has no more than 4 candidates, 2 of which pair into cells from units on opposite ends, this has become a naked double loop. Be careful, there are many semblances and parts of naked double loops out there. If any of the cells has a single value that does not pair in to either side, or if the loop doesn't close back in on itself, it is not a naked double loop.
What does this mean? Well, let's follow the logic.
Consider the two cells in the second column in the bottom-left box. If one is a 4 and one is an 8, then we can eliminate the 4 and the 8 in green just above them. If neither is a 4 or an eight, then one is a 1 and the other a 6. Then, among the other two yellow cells in that box one must be a 4 and the other a 5. If we follow that logic around the loop we see that among the two cells in the second column in the top-left box, one must be a 4 and the other an 8, and we can once again make the eliminations in green. Lastly, if only one is a 4 or an 8, then the other must be a 1 or 6. Now, among the other two yellow cells in that box, since now one can't be a 1 or a 6, then at least one must be a 4 or a 5. Then, on the far-right at least one must be a 2 or a 7. Following the logic around the loop we see that among the second column in the top-left box, at least one must be a 4 or an 8. Since we've already used a 4 or an 8 at the beginning directly below, then this must contain the other. Thus, between the top and bottom, we can yet again make the elimination between of the 4 and the 8. Thus, no matter what the circumstance, those two get eliminated.
If we change the starting point to any other pair of cells, we can see that all the values in green can be eliminated.
Once you get your mind around what a double loop looks like, you'll be able to spot them quicker that other techniques near this skill level. Simply look for cells that have 4 or fewer candidates, and have these connected pairs.
Since this technique is unique to Sudoku Snake and can not be researched elsewhere, we'll given you another example to see if you can spot the naked double loop. The example below is one of the most difficult examples of a naked double loop to spot because every cell in the puzzle has 4 or fewer candidates. Also note this - the naked double loop involves only two boxes, not four. See if you can find it, then scroll to the bottom of the page for the answer (try not to peek).
Sudoku Snake gives Naked Double Loops a skill value of 1500.
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