Advanced

Cage splitting

Large cages are hard because the combination list is long. Cage splitting does not shrink the cage. It cuts the list into the pieces that sit inside a single box, row, or column, and each piece has a much shorter menu of sums.

Imagine a five-cell cage summing to 23 that puts three cells in one box and two in the next. The three-cell side cannot be an arbitrary triple. It has to be a triple that leaves a legal pair for the other side, and both sides have to respect the digits already seen in their own boxes. Enumerate the full combinations once, group them by what they do to the three-cell side, and you often find that only two partial sums survive. Those two sums are a new constraint on that box, exactly as if a cage had been drawn around those three cells with an unknown total chosen from a very small set.

The practical version, which is enough for most hard puzzles, is visual. Color or trace the cells of one cage that share a box. Ask which digits on that side appear in every surviving combination. Those digits are locked to that side of the cage. If they also share a line inside the box, they point out of the box and delete themselves from the rest of the line. The split did not give you a number. It gave you a pointing pair, which is usually how the next single appears.

Splitting also works against the 45 rule. Once you know the three-cell side must sum to 6, 7, or 8, the innies of that box have a range instead of a single total. Ranges still delete. A cell that would have to be a 9 to finish the box can be removed if every legal partial sum leaves no room for a 9.

Do this only for cages that actually cross a border. A cage that lives inside one box is already split. And do it after the unique cages and the obvious 45 steps are exhausted. Splitting a cage that still has a naked single inside it is how players manufacture complexity the puzzle does not require.

If you want to practice the move on purpose, play a hard puzzle and, the first time you have no singles, pick the cage that crosses the most box borders. List the combinations that remain. The border is doing more work than the sum printed in the corner.

Use this technique on a puzzle

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