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Hidden cages

A hidden cage is a set of cells that must contain a known set of digits, or must add up to a known total, even though no dashed line was drawn around them. Killer puzzles are full of these, and they are the reason a blank-looking expert board is still logical.

The combination kind is the one to learn first. List every combination still legal for a cage. If the digit 5 appears in all of them, 5 is inside the cage no matter which combination is right. You can delete 5 from every cell outside the cage that sees all the cells where 5 could still go. When those possible cells lie in one row, the deletion runs along the whole row. The cage has grown a pointing constraint it never printed.

The arithmetic kind comes from the 45 rule. Two innies adding to 11 are a hidden cage of sum 11. They deserve the same combination list as a printed cage. Write the sum next to them in pencil if you need to. Forgetting a discovered sum is the usual reason a later step looks impossible.

Hidden cages also form when a house is almost filled by real cages. The two cells left over are a cage whether you noticed or not. Their sum is 45 minus the cages you can already add. Players who only inspect printed corners walk past these every game. Make one pass per box that asks “what is left?” rather than “what is marked?”

On the hardest boards a hidden cage on one side of a split meets a hidden cage from the 45 rule on the other, and the two lists have only one combination in common. That coincidence is not a guess. It is two independent constraints selecting the same digits. The hint text will name both ideas when that is the simplest step.

Treat every discovered sum as ink. If you would not ignore a printed cage, do not ignore one you calculated. The puzzles in our hard and expert sets are built so that these steps repeat, rather than appearing once and then abandoning you to trial and error.

Use this technique on a puzzle

Keep the combination table open while you read, and the rules if the cages themselves are still new.