Killer Sudoku Cage Combinations

Killer Sudoku cage combinations are the short table every Killer solver keeps in their head: the cage sums that can be made in exactly one way. Spot one of those sums and you know the digits before you have written a single candidate. It is the first habit of killer cage sudoku solving.

What the pattern is

A cage is a group of cells with a target sum, and no digit may repeat inside it. Most sums split several ways, but at the extremes there is only one option. Three different digits adding to 6 can only be 1, 2 and 3, because nothing else fits into so small a total.

When a cage has just one combination, those digits are locked into it and every other candidate leaves its cells. If every cell of the cage also sits in one unit, the digits can leave the rest of that unit too, exactly as a naked set would.

The forced sums

For every cage size and sum, see the full combinations chart.

Five-cell cages behave the same way at 15, 16, 34 and 35. The pattern is easy to remember: the lowest possible total for a cage of any size uses the smallest digits, the highest uses the largest, and the sums one step in from each end have exactly one combination as well.

When to look for it

• On your first pass, before any candidate marking. These sums are read off the printed clues.
• On the smallest and largest sums on the grid, which is where single combinations live.
• On every two-cell cage, since two-cell sums are the most restrictive on the board.
• Again later, once eliminations have knocked digits out of a cage and cut its options down to one.

How to apply it, step by step

1. Sweep the grid for cages of two, three and four cells.
2. Check each sum against the minimum and maximum for that cage size. The minimum is 1 plus 2 plus 3 and so on, the maximum uses the biggest digits.
3. If the sum matches a forced row above, write those digits into the cage cells as their only candidates.
4. Erase every other candidate from those cells.
5. If the cage lies wholly in one row, column or box, erase those digits from the rest of that unit as a separate step.
6. Re-check neighbouring cages and units, because a locked cage often exposes a single straight away.

Worked examples

The high end of the table works like the low end. Look at the 24 cage E7, F6, F7.

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Step 0 of 4

1. The puzzle has just started. Look at the 24 cage E7, F6, F7.

2. 1

   The 24 cage has three cells, and a cage never repeats a digit.

3. 2

   The three largest digits add up to 7 + 8 + 9 = 24, so that is the only set that fits.

4. 3

   Any digit below 7 in E7, F6 or F7 would leave the other two cells needing more than 8 + 9 = 17.

5. 4

   1, 3, 4, 5 and 6 removed from E7, F6 and F7, and 2 from F6, because the 24 cage can only hold 7, 8 and 9.

The puzzle has just started. Look at the 24 cage E7, F6, F7.

Three different digits reach 24 only as 7 + 8 + 9, the three largest digits. A digit below 7 in any of the three cells would leave the other two needing more than 8 + 9 = 17. So 1, 3, 4, 5 and 6 leave all three cells, and 2 leaves F6.

A two-cell cage works the same way. Look at the 16 cage A5, B5.

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Step 0 of 4

1. Easier techniques are used up. Look at the 16 cage A5, B5.

2. 1

   The 16 cage has two cells, and they cannot repeat a digit.

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   Two different digits reach 16 only as 7 + 9 (8 + 8 repeats a digit), so the cage holds exactly 7 and 9.

4. 3

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5. 4

   1 and 2 removed from A5 because the 16 cage can only hold 7 and 9.

Easier techniques are used up. Look at the 16 cage A5, B5.

Two different digits reach 16 only as 7 + 9, because 8 + 8 repeats a digit. A5 still carried 1 and 2, so both go. Erasing the digits from the rest of a unit is a separate step. It applies only when the cage lies wholly inside that unit, and it is a later move of its own.

Note what the combination does not tell you: it fixes the set of digits, never their order inside the cage.

Common mistakes

• Assuming positions. Knowing a cage is 1, 2 and 4 says nothing about which cell holds which digit until another rule decides.
• Allowing repeats. A two-cell cage of 4 is 1 and 3, not 2 and 2. Digits never repeat inside a cage.
• Eliminating outside a unit. The unit-wide erase only works for a unit that contains every cell of the cage. An L-shaped cage often shares a box but not a row.
• Stopping after the first pass. Cages with two or three combinations become forced later, as placements rule options out.

Where you meet it

Cage combinations are the opening move on easy Killer Sudoku and remain the backbone of every Killer solve above it. Once you are fluent with the forced sums, the next step is working with cages that keep several combinations and squeezing information out of what those combinations share.