Rule of 45
Every row, column and box holds the digits 1 to 9 exactly once, so its digits always add to 45. Cages that stick out reveal the difference.
What it is
Because each unit contains all nine digits, its total is fixed at 45. If the cages inside a unit add to less than 45, the shortfall belongs to cells reaching in from outside. If they add to more, the excess belongs to cells reaching out. Either way, one number is determined without knowing any individual digit.
How to spot it
Pick a row, column or box where cages mostly stay inside. Add the sums of the cages fully contained in it, then compare with 45. The difference is the total of the remaining cells.
Worked example
Step through the example. First the position, then the cage the argument rests on, then the candidates, and finally what follows from it.
Step 1 of 4: The starting position
- Pattern cells
- 1/1, 1/2, 1/3, 2/1, 2/2, 2/3, 3/1, 3/2, 3/3
- Digits
- 1
- Follows from it
- 3/3
Press a marked square to point it out on the board above.
Why it matters
The digits 1 through 9 add to 45, and every unit contains each exactly once. Cage sums are just a different partition of the same cells, so any mismatch with 45 is accounted for by the cells that cross the boundary.
Practise now
The practice puzzle is picked so that this technique is what gets the solve moving again.
Frequently asked questions
Why exactly 45?
Because 1+2+3+4+5+6+7+8+9 = 45, and every row, column and box contains each of those digits once.
Does it work for several units at once?
Yes. Two rows total 90, three total 135. Combining units is how the harder Killer deductions get made.
When does the rule help most?
When exactly one cell sticks out of a unit. Then the difference from 45 is that cell's digit, directly.
Put it to use
Where you will meet it
Hard grids are built around moves like this one. That is where it earns its keep.
Play HardStuck on a grid of your own?
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