The Mathematics of Sudoku II – Tom Davis –

13 Apr
3 Locked Candidates Locked candidates are forced to be within a certain part of a row, column or block. Sometimes you can find a block where the only possible positions for a candidate are in one row or column within that block. Since the block must contain the candidate, the candidate must appear in that row or column within the block. (在一个格子里,寻找它的行,锁定它必须为某个值,该值在该格子里只出现在该行,这样该行其它的这个值就会被剔除;在一个格子里,寻找它的列,锁定它必须为某个值,该值在该格子里只出现在该列,这样该列其它的这个值就会被剔除。功力在提升……)This means that you can eliminate the candidate as a possibility in the intersection of that row or column with other blocks. <<<< Figure 3: Locked Candidates—————————- A similar situation can occur when a number missing from a row or column can occur only within one of the blocks that intersect that row or column. Thus the candidate must lie on the intersection of the row/column and block and hence cannot be a candidate in any of the other squares that make up the block. Both of these situations are illustrated in Figure 3. The block def789 must contain a 2, and the only places this can occur are in squares f7 and f8: both in row f. Therefore 2 cannot be a candidate in any other squares in row f, including square f5 (so f5 must contain a 3). Similarly, the 2 in block ghi456 must lie in column 4 so 2 cannot be a candidate in any other squares of that column, including d4. Finally, the 5 that must occur in column 9 has to fall within the block def789 so 5 cannot be a candidate in any of the other squares in block def789, including f7 and f8. 4 Naked and Hidden Pairs, Triplets, Quads, . . . These are similar to naked singles, discussed in Section 2.2, except that instead of having only one candidate in a cell, you have the same two candidates in two cells (or, in the case of naked triplets, the same three candidates in three cells, et cetera).( 对于这一点还是非常受用的,二、三、四重奏的分辨很频繁,但三重奏、四重奏的变体需要一些时间来理解和灵活运用。一个小技巧——记住你的着眼点,是三个小格子或者四个小格子 in one virtual line,这样在分辨三重奏或者四重奏的变体处理会很容易分辨出来。) A naked pair, triplet or quad must be in the same virtual line. A naked triplet’s three values must be the only values that occur in three squares (and similarly, a naked quad’s four values must be the only ones occurring in four squares). When this occurs, those n squares must contain all and only those n values, where n = 1, 2 or 3. Those values can be eliminated as candidates from any other square in that virtual line. Figure 4 shows how to use a naked pair. In squares a2 and a8 the only candidates that appear are a 2 and a 7. That means that 7 must be in one, and 2 in the other. But then the 2 and 7 cannot appear in any of the other squares in that row, so 2 can be eliminated as a candidate in a3 and both 2 and 7 can be eliminated as candidates in a9. Figure 4: A Naked Pair For a naked pair, both squares must have exactly the same two candidates, but for naked triplets, quads, et cetera, the only requirement is that the three (or four) values be the only values appearing in those squares in some virtual line. For example, if three entries in a row admit the following sets of candidates: {1, 3}, {3, 7} and {1, 7} then it is impossible for a 1, 3 or 7 to appear in any other square of that row. ———————————-Figure 5: A Naked Triple >>>> Figure 5 contains a naked triple. In row a squares a2, a8 and a9 contain the naked triple consisting of the numbers 1, 3 and 7. Thus those numbers must appear in those squares in some order. For that reason, 1 and 3 can be eliminated as candidates from squares a4 and a5. Hidden pairs, triples and quads are related to naked pairs, triples and quads in the same way that hidden singles are related to naked singles. In Figure 6 consider row i. The only squares in row i in which the values 1, 4 and 8 appear are in squares i1, i5 and i6. Therefore we can eliminate candidates 2 and 6 from square i1 and candidate 3 from i5. ——————————–Figure 6: A Hidden Triple >>>> Remember, of course, that although the three examples above illustrate the naked and hidden sets in a row, these sets can appear in any virtual line: a row, column, or block. There is also no reason that there could not be a naked or hidden quintet, sextet, and so on, especially for versions of sudoku on grids that are larger than 9 × 9.
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Posted by on 04/13/2008 in ENTERTAINMENT


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