Killer Sudoku / solving lessons

Killer Sudoku: Understanding Cages and Sums

If you've played regular Sudoku, you already know most of what you need to play Killer Sudoku.

The difference is those little dotted boxes with numbers in the corners. They're called cages, and they can help you solve a puzzle before you've placed your first number.

What changes in Killer Sudoku?

The normal Sudoku rules still apply. Every row, column, and 3×3 box must contain the numbers 1 through 9 without repeating any.

Killer Sudoku adds two more rules:

  • The numbers inside each dotted cage must add up to the small total shown in its corner.
  • Numbers cannot repeat within the same cage, even when its cells are in different rows or boxes.

Those two rules give you another way to work out where numbers belong.

Start with the easiest cages

Imagine a cage containing just two squares. Its total is 3. What two different numbers between 1 and 9 can add up to 3?

Try the two-square cage totaling 17, too.

Two blank two-cell cages. Their totals are 3 and 17.
Each pair of squares is one cage. The small number is its total.
Show the combinationsThe 3 cage contains 1 and 2 in either order. The 17 cage contains 8 and 9 in either order.

Only 1 and 2 make 3. Only 8 and 9 make 17. In both cages, either order is still possible.

Not every cage has just one combination. A two-square cage totaling 10 could contain 1 and 9, 2 and 8, 3 and 7, or 4 and 6. You need another clue to narrow it down.

Use regular Sudoku rules to place the digits

Now put the 3 cage vertically. We know its digits are 1 and 2. A 1 is already visible in the top square's row. Which digit goes on top?

Two-row grid crop with four complete cages. The left vertical cage totals 3. A visible 1 is in its top row at the right.
The 1 at right is in the same row as the top square of the left cage.
Show where the digits goThe left cage has 2 above 1. The given 1 remains in the top row at the right.

The top square cannot be another 1, so it must be 2. The bottom square is 1. The cage gave us the pair. The row rule fixed their positions.

Every row, column, and box adds up to 45

Every complete row contains 1 through 9. Add them together and you get 45. The same is true for every column and every 3×3 box.

Here, three cages sit completely inside one box. They cover eight of its nine squares and total 5, 10, and 24. What must go in the ninth square?

A 3 by 3 box has internal cages of 5, 10, and 24 covering eight cells. The remaining bottom-left cell is part of an 8 cage that extends below the box.
The cage around the bottom-left square continues below the 3×3 box.
Show the missing digitThe bottom-left cell of the 3 by 3 box is 6. It belongs to the cage of 8 extending below the box.

The three cages total 5 + 10 + 24 = 39. The box totals 45, so the remaining square is 45 − 39 = 6.

Watch the boundary when you use this trick. If a cage crosses out of the row, column, or box you're examining, you cannot count its whole total as though every cell were inside.

What should you look for first?

Start with small cages, especially ones with unusually low or high totals. A two-square cage totaling 3 or 17 is a good place to begin. Then check the row, column, and box rules to see where those digits can go.

If you get stuck, look for a row, column, or box where cage totals make the rule of 45 useful.

Knowing which numbers belong in a cage does not always tell you their positions. That's still progress. A few eliminated possibilities can unlock another part of the puzzle.

Give it a try

Killer Sudoku uses the same logic as regular Sudoku, with cage totals giving you extra clues. The next time a puzzle looks crowded with dotted lines, pick one small cage and see what it tells you.

Try it in Daily Variant Sudoku, where Killer Sudoku is the current variant.