CageDokuAdvanced Killer Sudoku Techniques
By Andrew Boyko, developer of CageDoku · Updated
Once unique cages and single innies and outies no longer finish the puzzle, these techniques take over. They are what separate Moderate from Diabolical and Extreme.
Multi-region 45
The 45 rule is not limited to one row, column or box. Two adjacent rows add up to 90, three rows (a band of boxes) to 135, and the same holds for columns. Choose a set of regions, add every cage that lies completely inside it, and look at what is left. Two rows plus the cages inside them, for example, often leave exactly one or two cells that you can pin down by subtraction.
Pairs of innies and outies
If two cells stick out of a region, you know their sum even if you cannot name them. Combine that sum with the cage combinations: a pair sum of 3 is 1+2, 4 is 1+3, 16 is 7+9 and 17 is 8+9. A pair sum of 10 allows 1+9, 2+8, 3+7 and 4+6 but never 5+5, because digits cannot repeat when the two cells see each other.
Cage-line interaction
When a cage lies entirely inside one row, column or box, every digit in the cage is removed from the rest of that unit. If a combination must contain a digit, that digit can be erased from the other cells of the unit; if the cage can only use certain digits, the others may be free for the rest. This is the Killer version of locked candidates.
Combination clashes
Two cages that share a row, column or box cannot use the same digit. Suppose a 2-cell cage with sum 17 (which must be 8 and 9) sits in a row, and a 3-cell cage with sum 18 lies in the same row. Seven combinations make 18: 1-8-9, 2-7-9, 3-6-9, 3-7-8, 4-5-9, 4-6-8 and 5-6-7. Only 5-6-7 avoids 8 and 9, so the cage must be 5-6-7. Whenever a unit already holds a forced set of digits, strike every combination of the other cages that needs them.
Hidden and naked pairs inside cages
Two cells in a cage with the same two candidates form a naked pair; they lock those two digits and remove them from the rest of the cage and any shared unit. Hidden pairs work the same way when two digits can only go in two cells.
Fish and chains
The hardest puzzles in CageDoku (Extreme) can ask for X-Wing, Swordfish or a simple chain. If a hint shows one of these, follow the highlighted cells: seeing a pattern explained once makes it far easier to spot the next time.
Practice on real puzzles
Techniques stick when you meet them in a real grid. Try a Tough or Diabolical puzzle in your browser, and use hints only to learn the name of the technique that you missed.
Frequently asked questions
Which is harder: a 3-region 45 or a pair of outies?
Pairs are usually easier to spot once you have the sums; larger regions give more arithmetic but fewer stray cells.
When do I need fish or chains?
Only on the hardest puzzles. If you are on Gentle to Tough, the techniques above are enough.
Practice it in CageDoku
Free, ad-free Killer Sudoku and seven more variants. Hints name the technique and highlight the cells, so you learn why a digit goes where it goes.