Killer sudoku
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Killer sudoku (also killer su doku, sumdoku, sum doku, addoku, or samunamupure) is a puzzle that combines elements of sudoku and kakuro. Despite the name, the simpler killer sudokus can be easier to solve than regular sudokus, depending on the solver's skill at mental arithmetic; the hardest ones, however, can take hours to crack.
A typical problem is shown on the right, using colors to define the groups of cells. More usually, puzzles are printed in black and white, with thin dotted lines used to outline the "cages" (see below for terminology).
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[edit] History
Killer sudokus were already an established variant of sudoku in Japan by the mid 1990s, where they were known as "samunamupure." The name stemmed from a Japanized form of the English words "sum number place." Killer sudokus were introduced to most of the English-speaking world by The Times in 2005.
Traditionally, as with regular sudoku puzzles, the grid layout is symmetrical around a diagonal, horizontal or vertical axis. This is a matter of aesthetics, though, rather than obligatory: many Japanese puzzle-makers will make small deviations from perfect symmetry for the sake of improving the puzzle. Other puzzle-makers may produce entirely asymmetrical puzzles.
[edit] Terminology
- Cell
- A single square that contains one number in the grid
- Row
- A horizontal line of 9 cells
- Column
- A vertical line of 9 cells
- Nonet
- A 3×3 grid of cells, as outlined by the bolder lines in the diagram above
- Cage
- The grouping of cells denoted by a dotted line or by individual colours.
- House
- Any nonrepeating set of 9 cells: can be used as a general term for "row, cell, or nonet" (or, in Killer X variants, "long diagonal")
[edit] Rules
The objective is to fill the grid with numbers from 1 to 9 in a way that the following conditions are met:
- Each row, column, and nonet contains each number exactly once.
- The sum of all numbers in a cage must match the small number printed in its corner.
- No number appears more than once in a cage. (This is the standard rule for killer sudokus, and implies that no cage can include more than 9 cells.)
In 'Killer X', an additional rule is that each of the long diagonals contains each number once.
[edit] Duplicate cell ambiguity
By convention in Japan, killer sudoku cages do not include duplicate numbers. When The Times first introduced the killer sudoku in 2005, however, the newspaper did not make this rule explicit. Even though the vast majority of sudoku puzzles followed the rule anyway, English-speaking solvers scratched their heads over appropriate solving strategies given the ambiguity. Within a month, though, the Times clarified the rule, and now the world standard is no duplicates within cages.
[edit] Solving strategies
[edit] Fewest possible combinations
Generally the problem is best tackled starting from the extreme sums — cages with the largest or the smallest sums. This is because these have the fewest possible combinations. For example, 3 cells totalling 23 can only be 6, 8, and 9.
In the early stages of the game, the most common way to begin filling in numbers is to look at such low-sum or high-sum cages that form a 'straight line'. As the solver can infer from these that certain numbers are in a certain row or column, he can begin 'cross-hatching' across from them.
[edit] The 45 rule
A further technique can be derived from the knowledge that the numbers in all houses (rows, columns and nonets) add up to 45. By adding up the cages and single numbers in a particular house, the user can deduce the result of a single cell. If the cell calculated is within the house itself, it is referred to as an 'innie'; conversely if the cell is outside it, it is called an 'outie'. Even if this is not possible, advanced players may find it useful to derive the sum of two or three cells, then use other elimination techniques (see below for an example of this). The '45' technique can also be extended to calculate the innies or outies of N adjacent houses, as the difference between the cage-sums and N*45.
[edit] Initial analysis of the sample problem
[edit] Fewest possible combinations
The two cells in the top left must be 1+2. The 3 cells to the right totaling 15 cannot therefore have either a 1 or a 2, so they must be either 3+4+8, 3+5+7, or 4+5+6.
The two vertical cells in the top left of the top right nonet cannot be 2+2 as that would mean duplicates, so they must be 1+3. The 1 cannot be in the top line as that conflicts with our first 2 cells therefore the top cell of this pair is 3 and the lower cell 1. This also means the 3 cell cage 15 to the left cannot contain a 3 and so is 4+5+6.
Similarly the neighbouring 16 must be 9+7.
The four cells in the top right cage (totaling 15) can only include one of 1, 3, 7, or 9 (if at all) because of the presence of 1, 3, 7, and 9 in the top right hand nonet. If any one of 1, 3, 7, or 9 is present then this must be the lone square in the nonet below. Therefore these 4 cells is one of 1+2+4+8 or 2+3+4+6.
The 2 cells in the middle of the left edge must be either 1+5 or 2+4. And so on.
[edit] 45
Looking at the nonet on the left hand side in the middle, we can see that there are three cages which do not cross over into another nonet; these add up to 33, meaning that the sum of the remaining two cells must be 12. This does not seem particularly useful, but consider that the cell in the bottom right of the nonet is part of a 3-cage of 6; it can therefore only contain 1, 2 or 3. If it contained 1 or 2, the other cell would have to contain 11 or 10 respectively; this is impossible. It must, therefore, contain 3, and the other cell 9.
[edit] Cage total tables
The following tables list the possible combinations for various sums.
[edit] 2 cell
03: 12 04: 13 05: 14 23 06: 15 24 07: 16 25 34 08: 17 26 35 09: 18 27 36 45 10: 19 28 37 46 11: 29 38 47 56 12: 39 48 57 13: 49 58 67 14: 59 68 15: 69 78 16: 79 17: 89
[edit] 3 cell
06: 123 07: 124 08: 125 134 09: 126 135 234 10: 127 136 145 235 11: 128 137 146 236 245 12: 129 138 147 156 237 246 345 13: 139 148 157 238 247 256 346 14: 149 158 167 239 248 257 347 356 15: 159 168 249 258 267 348 357 456 16: 169 178 259 268 349 358 367 457 17: 179 269 278 359 368 458 467 18: 189 279 369 378 459 468 567 19: 289 379 469 478 568 20: 389 479 569 578 21: 489 579 678 22: 589 679 23: 689 24: 789
[edit] 4 cell
10: 1234 11: 1235 12: 1236 1245 13: 1237 1246 1345 14: 1238 1247 1256 1346 2345 15: 1239 1248 1257 1347 1356 2346 16: 1249 1258 1267 1348 1357 1456 2347 2356 17: 1259 1268 1349 1358 1367 1457 2348 2357 2456 18: 1269 1278 1359 1368 1458 1467 2349 2358 2367 2457 3456 19: 1279 1369 1378 1459 1468 1567 2359 2368 2458 2467 3457 20: 1289 1379 1469 1478 1568 2369 2378 2459 2468 2567 3458 3467 21: 1389 1479 1569 1578 2379 2469 2478 2568 3459 3468 3567 22: 1489 1579 1678 2389 2479 2569 2578 3469 3478 3568 4567 23: 1589 1679 2489 2579 2678 3479 3569 3578 4568 24: 1689 2589 2679 3489 3579 3678 4569 4578 25: 1789 2689 3589 3679 4579 4678 26: 2789 3689 4589 4679 5678 27: 3789 4689 5679 28: 4789 5689 29: 5789 30: 6789
[edit] 5 cell
15: 12345 16: 12346 17: 12347 12356 18: 12348 12357 12456 19: 12349 12358 12367 12457 13456 20: 12359 12368 12458 12467 13457 23456 21: 12369 12378 12459 12468 12567 13458 13467 23457 22: 12379 12469 12478 12568 13459 13468 13567 23458 23467 23: 12389 12479 12569 12578 13469 13478 13568 14567 23459 23468 23567 24: 12489 12579 12678 13479 13569 13578 14568 23469 23478 23568 24567 25: 12589 12679 13489 13579 13678 14569 14578 23479 23569 23578 24568 34567 26: 12689 13589 13679 14579 14678 23489 23579 23678 24569 24578 34568 27: 12789 13689 14589 14679 15678 23589 23679 24579 24678 34569 34578 28: 13789 14689 15679 23689 24589 24679 25678 34579 34678 29: 14789 15689 23789 24689 25679 34589 34679 35678 30: 15789 24789 25689 34689 35679 45678 31: 16789 25789 34789 35689 45679 32: 26789 35789 45689 33: 36789 45789 34: 46789 35: 56789
[edit] 6, 7, and 8 cells
It is easiest to determine the combinations within large cages by means of complements. The table for 6 cell cages is the complement of the 3 cell table adding up to "45 - listed value"; similarly the 7 cell table complements the 2 cell table. An 8-cell cage is of course missing only one digit (45 - the sum of the cage).
[edit] See also
[edit] External links
- Too good for Fiendish? Then try Killer Su Doku - article in The Times
